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Summer Assignment 2016-2017 Math Analysis Honors (PreCalculus Honors) Mr. Mercado Welcome to PreCalculus Honors. PreCalculus Honors is a very challenging course if you lack the basics skills in Algebra 1, Geometry, and Algebra 2. Your summer assignment reviews the basics for each course. You can do the assignment anytime during the summer. To receive credit for the assignment it must be done completely, correctly with work shown on college/wide rule or graph paper. The assignment will be due on the first day of class. The practice test will be due on the second day of class and there will be an exam based on the summer assignmen. Please email me if you have any questions, [email protected]. Is PreCalculus Honors the right course for you? Answer truthfully the following questions for the answer. Yes No □ □ □ □ □ □ Did you have any missing assignments in Algebra 2? Did you turn in any assignments late in Algebra 2? Is your test average for the year a C or lower in Algebra 2? If you answered yes to any of the above questions, PreCalculus Honors may not be the right course for you and may result with you not passing PreCalculus Honors. Mr. Mercado Mr. Mercado Summer Assignment (272 total questions) Geometry Chapter 1.7 Page 52 #1 – 33 all (33) Chapter 3.2 Page 157 #1 – 35 odd (18) Chapter 4.1 Page 221 #1 – 37 odd (19) Chapter 7.1 Page 436 #1 – 23 odd (12) Chapter 7.4 Page 461 #1 – 25 odd (13) Algebra 2 Chapter A1 Page A12 #11 – 67 odd (29) Chapter A2 Page A24 #11 – 115 e.o.o.* (27) Chapter A3 Page A35 #9 – 201 e.o.o.* (49) Chapter A4 Page A46 #7 – 83 e.o.o (20) Chapter A5 Page A60 #5 – 97 e.o.o.*, #127 – 143 odd (33) Chapter A6 Page A69 #7 – 79 e.o.o. (19) *e.o.o. stands for every other odd, which means to do every second odd or keep adding 4; Examples of e.o.o.: #1, 5, 9, 13, 17, 21, 25, 29, etc. #5, 9, 13, 17, etc. #9, 13, 17, 21, etc. #11, 15, 19, 23, etc. Before Now Why? Key Vocabulary • perimeter, p. 923 • circumference, p. 923 • area, p. 923 • diameter, p. 923 • radius, p. 923 You classified polygons. You will find dimensions of polygons. So you can use measures in science, as in Ex. 46. Recall that perimeter is the distance around a figure, circumference is the distance around a circle, and area is the amount of surface covered by a figure. Perimeter and circumference are measured in units of length, such as meters (m) and feet (ft). Area is measured in square units, such as square meters (m 2) and square feet (ft 2). For Your Notebook KEY CONCEPT Formulas for Perimeter P, Area A, and Circumference C Square Rectangle side length s P 5 4s A 5 s2 length l and width w P 5 2l 1 2w A 5 lw s Triangle side lengths a, b, and c, base b, and height h P5a1b1c a r diameter d and radius r C 5 πd 5 2πr A 5 πr 2 b 1 A5} bh d Pi (p ) is the ratio of a circle’s circumference to its diameter. 2 EXAMPLE 1 w Circle c h l Find the perimeter and area of a rectangle FT BASKETBALL Find the perimeter and area of the rectangular basketball court shown. Perimeter Area P 5 2l 1 2w A 5 lw 5 2(84) 1 2(50) 5 84(50) 5 268 5 4200 FT 1.7 Find Perimeter, Circumference, and Area c The perimeter is 268 feet and the area is 4200 square feet. 1.7 Find Perimeter, Circumference, and Area 49 EXAMPLE 2 Find the circumference and area of a circle TEAM PATCH You are ordering circular cloth patches for your soccer team’s uniforms. Find the approximate circumference and area of the patch shown. Solution APPROXIMATE p First find the radius. The diameter is 9 centimeters, The approximations 1 so the radius is } (9) 5 4.5 centimeters. 2 22 3.14 and } are 7 commonly used as approximations for the irrational number π. Unless told otherwise, use 3.14 for π. ✓ CM Then find the circumference and area. Use 3.14 to approximate the value of π. C 5 2πr ø 2(3.14)(4.5) 5 28.26 A 5 πr 2 ø 3.14(4.5)2 5 63.585 c The circumference is about 28.3 cm. The area is about 63.6 cm2. GUIDED PRACTICE for Examples 1 and 2 Find the area and perimeter (or circumference) of the figure. If necessary, round to the nearest tenth. 1. 2. 3. 5.7 m 2 yd 1.6 cm 13 m ★ EXAMPLE 3 Standardized Test Practice Triangle QRS has vertices Q(1, 2), R(4, 6), and S(5, 2). What is the approximate perimeter of triangle QRS? A 8 units B 8.3 units C 13.1 units D 25.4 units Solution AVOID ERRORS Write down your calculations to make sure you do not make a mistake substituting values in the Distance Formula. First draw triangle QRS in a coordinate plane. Find the side lengths. Use the Distance Formula to find QR and RS. R(4, 6) QS 5 ⏐5 2 1⏐ 5 4 units }} } }} } QR 5 Ï(4 2 1)2 1 (6 2 2)2 5 Ï25 5 5 units RS 5 Ï(5 2 4)2 1 (2 2 6)2 5 Ï17 ø 4.1 units P 5 QS 1 QR 1 RS ø 4 1 5 1 4.1 5 13.1 units c The correct answer is C. Chapter 1 Essentials of Geometry 1 Œ(1, 2) 1 Then find the perimeter. 50 y A B C D S(5, 2) x EXAMPLE 4 Solve a multi-step problem SKATING RINK An ice-resurfacing machine is used to smooth the surface of the ice at a skating rink. The machine can resurface about 270 square yards of ice in one minute. About how many minutes does it take the machine to resurface a rectangular skating rink that is 200 feet long and 90 feet wide? ANOTHER WAY Solution For an alternative method for solving the problem in Example 4, turn to page 57 for the Problem Solving Workshop. The machine can resurface the ice at a rate of 270 square yards per minute. So, the amount of time it takes to resurface the skating rink depends on its area. STEP 1 Find the area of the rectangular skating rink. Area 5 lw 5 200(90) 5 18,000 ft 2 The resurfacing rate is in square yards per minute. Rewrite the area of the rink in square yards. There are 3 feet in 1 yard, and 32 5 9 square feet in 1 square yard. 1 yd2 18,000 ft 2 p } 5 2000 yd 2 2 9 ft Use unit analysis. STEP 2 Write a verbal model to represent the situation. Then write and solve an equation based on the verbal model. Let t represent the total time (in minutes) needed to resurface the skating rink. Area of rink (yd2) 5 Resurfacing rate (yd2 per min) 2000 5 270 p t 3 Total time (min) Substitute. 7.4 ø t Divide each side by 270. c It takes the ice-resurfacing machine about 7 minutes to resurface the skating rink. ✓ GUIDED PRACTICE for Examples 3 and 4 4. Describe how to find the height from F y to } EG in the triangle at the right. E(1, 6) 5. Find the perimeter and the area of the triangle shown at the right. F(7, 3) 6. WHAT IF? In Example 4, suppose the skating rink is twice as long and twice as wide. Will it take an ice-resurfacing machine twice as long to resurface the skating rink? Explain your reasoning. 1 G(1, 2) 1 1.7 Find Perimeter, Circumference, and Area x 51 EXAMPLE 5 Find unknown length The base of a triangle is 28 meters. Its area is 308 square meters. Find the height of the triangle. h Solution 1 A5} bh Write formula for the area of a triangle. 2 1 308 5 } (28)h 2 28 m Substitute 308 for A and 28 for b. 22 5 h Solve for h. c The height is 22 meters. ✓ GUIDED PRACTICE for Example 5 7. The area of a triangle is 64 square meters, and its height is 16 meters. Find the length of its base. 1.7 EXERCISES HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 7, 21, and 41 ★ 5 STANDARDIZED TEST PRACTICE Exs. 2, 19, 26, 38, and 45 5 MULTIPLE REPRESENTATIONS Ex. 44 SKILL PRACTICE 1. VOCABULARY How are the diameter and radius of a circle related? 2. ★ WRITING Describe a real-world situation in which you would need to find a perimeter, and a situation in which you would need to find an area. What measurement units would you use in each situation? EXAMPLE 1 3. ERROR ANALYSIS Describe and correct the on p. 49 for Exs. 3–10 error made in finding the area of a triangle with a height of 9 feet and a base of 52 feet. A 5 52(9) 5 468 ft2 PERIMETER AND AREA Find the perimeter and area of the shaded figure. 4. 5. 6. 8 ft 4.2 m 18 ft 7. 30 yd 78 yd 7m 8. 9 mm 72 yd (FPNFUSZ 52 Chapter 1 Essentials of Geometry 15 in. 24 mm at classzone.com 15 mm 9. 10 cm 17 cm 9 cm 8 cm 6 cm 10. DRAWING A DIAGRAM The base of a triangle is 32 feet. Its height is 1 16 } feet. Sketch the triangle and find its area. 2 EXAMPLE 2 CIRCUMFERENCE AND AREA Use the given diameter d or radius r to find the on p. 50 for Exs. 11–15 circumference and area of the circle. Round to the nearest tenth. 11. d 5 27 cm 12. d 5 5 in. 13. r 5 12.1 cm 14. r 5 3.9 cm 15. DRAWING A DIAGRAM The diameter of a circle is 18.9 centimeters. Sketch the circle and find its circumference and area. Round your answers to the nearest tenth. EXAMPLE 3 DISTANCE FORMULA Find the perimeter of the figure. Round to the nearest on p. 50 for Exs. 16–19 tenth of a unit. 16. y 17. R 18. y G 2 M y N S 1 1 T 19. E x 1 x 1 F Œ x 1 ★ MULTIPLE CHOICE What is the approximate area (in square units) of the rectangle shown at the right? A A 6.7 B 8.0 1 C 9.0 D 10.0 P y B C D x 3 EXAMPLE 4 CONVERTING UNITS Copy and complete the statement. on p. 51 for Exs. 20–26 20 . 187 cm 2 5 ? m 2 21. 13 ft 2 5 ? yd 2 22. 18 in.2 5 ? ft 2 23. 8 km 2 5 ? m 2 24. 12 yd 2 5 ? ft 2 25. 24 ft 2 5 ? in.2 26. ★ MULTIPLE CHOICE A triangle has an area of 2.25 square feet. What is the area of the triangle in square inches? A 27 in.2 B 54 in.2 C 144 in.2 D 324 in.2 EXAMPLE 5 UNKNOWN MEASURES Use the information about the figure to find the on p. 52 for Exs. 27–30 indicated measure. 27. Area 5 261 m 2 Find the height h. 28. Area 5 66 in.2 29. Perimeter 5 25 in. Find the base b. 8 14 h Find the width w. in. w 36 m b 8 in. 1.7 Find Perimeter, Circumference, and Area 53 30. UNKNOWN MEASURE The width of a rectangle is 17 inches. Its perimeter is 102 inches. Find the length of the rectangle. 31. ALGEBRA The area of a rectangle is 18 square inches. The length of the rectangle is twice its width. Find the length and width of the rectangle. 32. ALGEBRA The area of a triangle is 27 square feet. Its height is three times the length of its base. Find the height and base of the triangle. 33. ALGEBRA Let x represent the side length of a square. Find a regular polygon with side length x whose perimeter is twice the perimeter of the square. Find a regular polygon with side length x whose perimeter is three times the length of the square. Explain your thinking. FINDING SIDE LENGTHS Find the side length of the square with the given area. Write your answer as a radical in simplest form. 34. A 5 184 cm 2 38. 35. A 5 346 in.2 36. A 5 1008 mi 2 37. A 5 1050 km 2 ★ SHORT RESPONSE In the diagram, the diameter of the yellow circle is half the diameter of the red circle. What fraction of the area of the red circle is not covered by the yellow circle? Explain. 39. CHALLENGE The area of a rectangle is 30 cm 2 and its perimeter is 26 cm. Find the length and width of the rectangle. PROBLEM SOLVING EXAMPLES 1 and 2 on pp. 49–50 for Exs. 40–41 40. WATER LILIES The giant Amazon water lily has a lily pad that is shaped like a circle. Find the circumference and area of a lily pad with a diameter of 60 inches. Round your answers to the nearest tenth. GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN 41. LAND You are planting grass on a rectangular plot of land. You are also building a fence around the edge of the plot. The plot is 45 yards long and 30 yards wide. How much area do you need to cover with grass seed? How many feet of fencing do you need? GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN EXAMPLE 4 on p. 51 for Ex. 42 42. MULTI-STEP PROBLEM Chris is installing a solar panel. The maximum amount of power the solar panel can generate in a day depends in part on its area. On a sunny day in the city where Chris lives, each square meter of the panel can generate up to 125 watts of power. The flat rectangular panel is 84 centimeters long and 54 centimeters wide. a. Find the area of the solar panel in square meters. b. What is the maximum amount of power (in watts) that the panel could generate if its area was 1 square meter? 2 square meters? Explain. c. Estimate the maximum amount of power Chris’s solar panel can generate. Explain your reasoning. 54 5 WORKED-OUT SOLUTIONS on p. WS1 ★ 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS 3.2 Before Now Why? Key Vocabulary • corresponding angles, p. 149 • alternate interior angles, p. 149 • alternate exterior angles, p. 149 • consecutive interior angles, p. 149 Use Parallel Lines and Transversals You identified angle pairs formed by a transversal. You will use angles formed by parallel lines and transversals. So you can understand angles formed by light, as in Example 4. ACTIVITY EXPLORE PARALLEL LINES Materials: lined paper, tracing paper, straightedge STEP 1 Draw a pair of parallel lines cut by a nonperpendicular transversal on lined paper. Label the angles as shown. 1 2 3 4 STEP 2 Trace your drawing onto tracing paper. STEP 3 Move the tracing paper to position ∠ 1 of the traced figure over ∠ 5 of the original figure. Compare the angles. Are they congruent? 5 6 7 8 STEP 4 Compare the eight angles and list all the congruent pairs. What do you notice about the special angle pairs formed by the transversal? For Your Notebook POSTULATE POSTULATE 15 Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. t 2 p 6 q ∠2 > ∠6 EXAMPLE 1 Identify congruent angles The measure of three of the numbered angles is 1208. Identify the angles. Explain your reasoning. Solution 1208 2 3 4 5 6 7 8 By the Corresponding Angles Postulate, m∠ 5 5 1208. Using the Vertical Angles Congruence Theorem, m∠ 4 5 1208. Because ∠ 4 and ∠ 8 are corresponding angles, by the Corresponding Angles Postulate, you know that m∠ 8 5 1208. 154 Chapter 3 Parallel and Perpendicular Lines For Your Notebook THEOREMS THEOREM 3.1 Alternate Interior Angles Theorem t If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. p 4 5 Proof: Example 3, p. 156 q ∠4 > ∠5 THEOREM 3.2 Alternate Exterior Angles Theorem t 1 If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. p q 8 Proof: Ex. 37, p. 159 ∠1 > ∠8 THEOREM 3.3 Consecutive Interior Angles Theorem t If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. q ∠ 3 and ∠ 5 are supplementary. Proof: Ex. 41, p. 159 EXAMPLE 2 p 3 5 Use properties of parallel lines ALGEBRA Find the value of x. 1158 4 (x 1 5)8 a b Solution By the Vertical Angles Congruence Theorem, m∠ 4 5 1158. Lines a and b are parallel, so you can use the theorems about parallel lines. m∠ 4 1 (x 1 5)8 5 1808 1158 1 (x 1 5)8 5 1808 x 1 120 5 180 x 5 60 (FPNFUSZ ✓ GUIDED PRACTICE Consecutive Interior Angles Theorem Substitute 115 8 for m ∠ 4. Combine like terms. Subtract 120 from each side. at classzone.com for Examples 1 and 2 Use the diagram at the right. 1. If m∠ 1 5 1058, find m∠ 4, m∠ 5, and m∠ 8. Tell which postulate or theorem you use in each case. 1 2 3 4 5 6 7 8 2. If m∠ 3 5 688 and m∠ 8 5 (2x 1 4)8, what is the value of x? Show your steps. 3.2 Use Parallel Lines and Transversals 155 EXAMPLE 3 Prove the Alternate Interior Angles Theorem Prove that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Solution WRITE PROOFS You can use the information from the diagram in your proof. Find any special angle pairs. Then decide what you know about those pairs. Draw a diagram. Label a pair of alternate interior angles as ∠ 1 and ∠ 2. You are looking for an angle that is related to both ∠ 1 and ∠ 2. Notice that one angle is a vertical angle with ∠ 2 and a corresponding angle with ∠ 1. Label it ∠ 3. GIVEN PROVE p 2 3 q cpiq c ∠1 > ∠2 STATEMENTS 1. 2. 3. 4. t 1 REASONS piq ∠1 > ∠3 ∠3 > ∠2 ∠1 > ∠2 EXAMPLE 4 1. 2. 3. 4. Given Corresponding Angles Postulate Vertical Angles Congruence Theorem Transitive Property of Congruence Solve a real-world problem SCIENCE When sunlight enters a drop of rain, different colors of light leave the drop at different angles. This process is what makes a rainbow. For violet light, m∠ 2 5 408. What is m∠ 1? How do you know? 2 1 Solution Because the sun’s rays are parallel, ∠ 1 and ∠ 2 are alternate interior angles. By the Alternate Interior Angles Theorem, ∠ 1 > ∠ 2. By the definition of congruent angles, m∠ 1 5 m∠ 2 5 408. ✓ GUIDED PRACTICE for Examples 3 and 4 3. In the proof in Example 3, if you use the third statement before the second statement, could you still prove the theorem? Explain. 4. WHAT IF? Suppose the diagram in Example 4 shows yellow light leaving a drop of rain. Yellow light leaves the drop at an angle of 418. What is m∠ 1 in this case? How do you know? 156 Chapter 3 Parallel and Perpendicular Lines 3.2 EXERCISES HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 5, 9, and 39 ★ 5 STANDARDIZED TEST PRACTICE Exs. 2, 3, 21, 33, 39, and 40 SKILL PRACTICE 1. VOCABULARY Draw a pair of parallel lines and a transversal. Label a pair of corresponding angles. EXAMPLES 1 and 2 2. ★ WRITING Two parallel lines are cut by a transversal. Which pairs of angles are congruent? Which pairs of angles are supplementary? 3. ★ on pp. 154–155 for Exs. 3–16 MULTIPLE CHOICE In the figure at the right, which angle has the same measure as ∠ 1? A ∠2 B ∠3 C ∠4 D ∠5 1 2 3 4 5 USING PARALLEL LINES Find the angle measure. Tell which postulate or theorem you use. 4. If m∠ 4 5 658, then m∠ 1 5 ? . 1 2 3 4 5 6 7 8 5. If m∠ 7 5 1108, then m∠ 2 5 ? . 6. If m∠ 5 5 718, then m∠ 4 5 ? . 7. If m∠ 3 5 1178, then m∠ 5 5 ? . 8. If m∠ 8 5 548, then m∠ 1 5 ? . USING POSTULATES AND THEOREMS What postulate or theorem justifies the statement about the diagram? 9. ∠ 1 > ∠ 5 10. ∠ 4 > ∠ 5 11. ∠ 2 > ∠ 7 12. ∠ 2 and ∠ 5 are supplementary. 13. ∠ 3 > ∠ 6 14. ∠ 3 > ∠ 7 15. ∠ 1 > ∠ 8 16. ∠ 4 and ∠ 7 are supplementary. 1 2 4 3 5 6 8 7 USING PARALLEL LINES Find m∠ 1 and m∠ 2. Explain your reasoning. 17. 18. 1508 19. 1408 1 2 2 1228 1 1 2 20. ERROR ANALYSIS A student concludes that ∠ 9 > ∠ 10 by the Corresponding Angles Postulate. Describe and correct the error in this reasoning. 9 ∠ 9 > ∠ 10 10 3.2 Use Parallel Lines and Transversals 157 21. SHORT RESPONSE Given p i q, describe two methods you can use to show that ∠ 1 > ∠ 4. t ★ 1 p 2 3 q 4 USING PARALLEL LINES Find m∠ 1, m∠ 2, and m∠ 3. Explain your reasoning. 22. 23. 1 24. 1 2 808 1 2 1338 3 1158 3 ANGLES Use the diagram at the right. 2 ‹]› ‹]› A 25. Name two pairs of congruent angles if AB and DC are parallel. ‹]› 3 B ‹]› 26. Name two pairs of supplementary angles if AD and BC D are parallel. C ALGEBRA Find the values of x and y. 27. 28. 29. 3y 8 y8 x8 6y 8 458 30. 31. 608 3x 8 33. ★ 32. 4x 8 (14x 2 10)8 (3y 1 2)8 (5y 2 5)8 2y 8 528 1358 B 75 C 110 D 115 t 1108 m (y 2 5)8 1 34. DRAWING Draw a four-sided figure with sides } MN and } PQ, such that }i } MN PQ, } MP i } NQ, and ∠ MNQ is an acute angle. Which angle pairs formed are congruent? Explain your reasoning. CHALLENGE Find the values of x and y. 35. 36. 608 1508 (5x 2 y)8 (5x 1 y)8 (2x 2 y)8 (2x 1 y)8 408 158 5x8 MULTIPLE CHOICE What is the value of y in the diagram? A 70 5 WORKED-OUT SOLUTIONS on p. WS1 658 558 2x 8 858 y8 x8 1308 ★ 5 STANDARDIZED TEST PRACTICE n 4.1 Before Now Why? Key Vocabulary • triangle scalene, isosceles, equilateral, acute, right, obtuse, equiangular • interior angles • exterior angles • corollary to a Apply Triangle Sum Properties You classified angles and found their measures. You will classify triangles and find measures of their angles. So you can place actors on stage, as in Ex. 40. A triangle is a polygon with three sides. A triangle with vertices A, B, and C is called “triangle ABC” or “n ABC.” For Your Notebook KEY CONCEPT Classifying Triangles by Sides Scalene Triangle Isosceles Triangle Equilateral Triangle No congruent sides At least 2 congruent sides 3 congruent sides theorem READ VOCABULARY Notice that an equilateral triangle is also isosceles. An equiangular triangle is also acute. Classifying Triangles by Angles Acute Triangle Right Triangle Obtuse Triangle Equiangular Triangle 3 acute angles 1 right angle 1 obtuse angle 3 congruent angles EXAMPLE 1 Classify triangles by sides and by angles SUPPORT BEAMS Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles. Solution The triangle has a pair of congruent sides, so it is isosceles. By measuring, the angles are 558, 558, and 708. It is an acute isosceles triangle. 4.1 Apply Triangle Sum Properties 217 EXAMPLE 2 Classify a triangle in a coordinate plane Classify nPQO by its sides. Then determine if the triangle is a right triangle. y Œ(6, 3) P(21, 2) O(0, 0) x Solution STEP 1 Use the distance formula to find the side lengths. }} }} }} }} }} }} OP 5 Ï (x2 2 x1)2 1 (y2 2 y1)2 5 Ï ((21) 2 0)2 1 (2 2 0)2 5 Ï5 ø 2.2 } } OQ 5 Ï (x2 2 x1)2 1 (y2 2 y1)2 5 Ï(6 2 0)2 1 (3 2 0)2 5 Ï45 ø 6.7 } PQ 5 Ï(x2 2 x1)2 1 (y2 2 y1)2 5 Ï (6 2 (21))2 1 (3 2 2)2 5 Ï 50 ø 7.1 220 STEP 2 Check for right angles. The slope of } OP is } 5 22. The slope 21 2 0 20 1 of } OQ is 3} 5} . The product of the slopes is 22 620 2 so } OP ⊥ } OQ and ∠ POQ is a right angle. 1 }12 2 5 21, c Therefore, nPQO is a right scalene triangle. ✓ GUIDED PRACTICE for Examples 1 and 2 1. Draw an obtuse isosceles triangle and an acute scalene triangle. 2. Triangle ABC has the vertices A(0, 0), B(3, 3), and C(23, 3). Classify it by its sides. Then determine if it is a right triangle. ANGLES When the sides of a polygon are extended, other angles are formed. The original angles are the interior angles. The angles that form linear pairs with the interior angles are the exterior angles. READ DIAGRAMS Each vertex has a pair of congruent exterior angles. However, it is common to show only one exterior angle at each vertex. interior angles exterior angles For Your Notebook THEOREM THEOREM 4.1 Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 1808. Proof: p. 219; Ex. 53, p. 224 218 Chapter 4 Congruent Triangles B A C m∠ A 1 m∠ B 1 m∠ C 5 1808 AUXILIARY LINES To prove certain theorems, you may need to add a line, a segment, or a ray to a given diagram. An auxiliary line is used in the proof of the Triangle Sum Theorem. PROOF GIVEN PROVE Triangle Sum Theorem c n ABC c m∠ 1 1 m∠ 2 1 m∠ 3 5 1808 B 4 D 2 5 Plan a. Draw an auxiliary line through 1 3 A C } for B and parallel to AC . Proof b. Show that m∠ 4 1 m∠ 2 1 m∠ 5 5 1808, ∠ 1 > ∠ 4, and ∠ 3 > ∠ 5. c. By substitution, m∠ 1 1 m∠ 2 1 m∠ 3 5 1808. STATEMENTS ‹]› Plan a. 1. Draw BD parallel to } AC. in Action b. 2. m∠ 4 1 m∠ 2 1 m∠ 5 5 1808 3. ∠ 1 > ∠ 4, ∠ 3 > ∠ 5 4. m∠ 1 5 m∠ 4, m∠ 3 5 m∠ 5 c. 5. m∠ 1 1 m∠ 2 1 m∠ 3 5 1808 REASONS 1. Parallel Postulate 2. Angle Addition Postulate and definition of straight angle 3. Alternate Interior Angles Theorem 4. Definition of congruent angles 5. Substitution Property of Equality For Your Notebook THEOREM THEOREM 4.2 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. B Proof: Ex. 50, p. 223 EXAMPLE 3 1 C A m∠ 1 5 m∠ A 1 m∠ B Find an angle measure J ALGEBRA Find m∠ JKM. x8 Solution STEP 1 Write and solve an equation to find the value of x. (2x 2 5)8 5 708 1 x8 x 5 75 708 L (2x 2 5)8 K M Apply the Exterior Angle Theorem. Solve for x. STEP 2 Substitute 75 for x in 2x 2 5 to find m∠ JKM. 2x 2 5 5 2 p 75 2 5 5 145 c The measure of ∠ JKM is 1458. 4.1 Apply Triangle Sum Properties 219 A corollary to a theorem is a statement that can be proved easily using the theorem. The corollary below follows from the Triangle Sum Theorem. For Your Notebook COROLLARY Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary. Proof: Ex. 48, p. 223 EXAMPLE 4 C A B m∠ A 1 m∠ B 5 908 Find angle measures from a verbal description ARCHITECTURE The tiled staircase shown forms a right triangle. The measure of one acute angle in the triangle is twice the measure of the other. Find the measure of each acute angle. Solution First, sketch a diagram of the situation. Let the measure of the smaller acute angle be x8. Then the measure of the larger acute angle is 2x8. The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary. 2x 8 x8 Use the corollary to set up and solve an equation. x8 1 2x8 5 908 x 5 30 Corollary to the Triangle Sum Theorem Solve for x. c So, the measures of the acute angles are 308 and 2(308) 5 608. ✓ GUIDED PRACTICE for Examples 3 and 4 3. Find the measure of ∠ 1 in the diagram 3x8 shown. 408 1 (5x 2 10)8 4. Find the measure of each interior angle of n ABC, where m∠ A 5 x8, m∠ B 5 2x8, and m∠ C 5 3x8. 5. Find the measures of the acute angles of the right triangle in the diagram shown. 2x8 (x 2 6)8 6. In Example 4, what is the measure of the obtuse angle formed between the staircase and a segment extending from the horizontal leg? 220 Chapter 4 Congruent Triangles 4.1 EXERCISES HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 9, 15, and 41 ★ 5 STANDARDIZED TEST PRACTICE Exs. 7, 20, 31, 43, and 51 SKILL PRACTICE VOCABULARY Match the triangle description with the most specific name. 1. Angle measures: 308, 608, 908 A. Isosceles 2. Side lengths: 2 cm, 2 cm, 2 cm B. Scalene 3. Angle measures: 608, 608, 608 C. Right 4. Side lengths: 6 m, 3 m, 6 m D. Obtuse 5. Side lengths: 5 ft, 7 ft, 9 ft E. Equilateral 6. Angle measures: 208, 1258, 358 F. Equiangular 7. ★ WRITING Can a right triangle also be obtuse? Explain why or why not. EXAMPLE 1 CLASSIFYING TRIANGLES Copy the triangle and measure its angles. Classify on p. 217 for Exs. 8–10 the triangle by its sides and by its angles. 8. X 9. Y EXAMPLE 2 on p. 218 for Exs. 11–13 Z 10. J M L K H N COORDINATE PLANE A triangle has the given vertices. Graph the triangle and classify it by its sides. Then determine if it is a right triangle. 11. A(2, 3), B(6, 3), C(2, 7) 12. A(3, 3), B(6, 9), C(6, 23) 13. A(1, 9), B(4, 8), C(2, 5) EXAMPLE 3 FINDING ANGLE MEASURES Find the value of x. Then classify the triangle by on p. 219 for Exs. 14–19 its angles. 14. 15. 608 x8 x8 608 16. 3x8 648 608 708 x8 ALGEBRA Find the measure of the exterior angle shown. 17. 18. 19. (2x 2 2)8 x8 EXAMPLE 4 on p. 220 for Ex. 20 20. 458 x8 248 (3x 1 6)8 (2x 1 18)8 1 (3x 1 2)8 ★ SHORT RESPONSE Explain how to use the Corollary to the Triangle Sum Theorem to find the measure of each angle. 3x8 2x8 4.1 Apply Triangle Sum Properties 221 ANGLE RELATIONSHIPS Find the measure of the numbered angle. 21. ∠ 1 22. ∠ 2 23. ∠ 3 24. ∠ 4 25. ∠ 5 26. ∠ 6 1 408 2 3 4 208 5 6 27. ALGEBRA In nPQR, ∠ P > ∠ R and the measure of ∠ Q is twice the measure of ∠ R. Find the measure of each angle. 28. ALGEBRA In nEFG, m∠ F 5 3(m∠ G), and m∠ E 5 m∠ F 2 308. Find the measure of each angle. ERROR ANALYSIS In Exercises 29 and 30, describe and correct the error. 29. 31. 30. All equilateral triangles are also isosceles. So, if n ABC is isosceles, then it is equilateral as well. ★ m∠ 1 1 808 1 508 5 1808 80° 1 50° MULTIPLE CHOICE Which of the following is not possible? A An acute scalene triangle B A triangle with two acute exterior angles C An obtuse isosceles triangle D An equiangular acute triangle ALGEBRA In Exercises 32–37, find the values of x and y. 32. 438 33. y8 34. 258 1188 758 x8 x8 y8 35. x8 228 y8 36. y8 37. y8 x8 x8 208 158 648 358 478 188 38. VISUALIZATION Is there an angle measure that is so small that any triangle with that angle measure will be an obtuse triangle? Explain. 39. CHALLENGE Suppose you have the equations y 5 ax 1 b, y 5 cx 1 d, and y 5 ex 1 f. a. When will these three lines form a triangle? b. Let c 5 1, d 5 2, e 5 4, and f 5 27. Find values of a and b so that no triangle is formed by the three equations. 4 3 4 1 41 c. Draw the triangle formed when a 5 } ,b5} , c 5 2}, d 5 } , e 5 0, 3 3 and f 5 21. Then classify the triangle by its sides. 222 5 WORKED-OUT SOLUTIONS on p. WS1 ★ 5 STANDARDIZED TEST PRACTICE 3 x8 y8 7.1 Before Apply the Pythagorean Theorem You learned about the relationships within triangles. Now You will find side lengths in right triangles. Why? So you can find the shortest distance to a campfire, as in Ex. 35. Key Vocabulary • Pythagorean triple • right triangle, p. 217 • leg of a right triangle, p. 241 • hypotenuse, p. 241 One of the most famous theorems in mathematics is the Pythagorean Theorem, named for the ancient Greek mathematician Pythagoras (around 500 B.C.). This theorem can be used to find information about the lengths of the sides of a right triangle. hypotenuse leg leg For Your Notebook THEOREM THEOREM 7.1 Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. b Proof: p. 434; Ex. 32, p. 455 EXAMPLE 1 c a c 2 5 a 2 1 b2 Find the length of a hypotenuse Find the length of the hypotenuse of the right triangle. 8 Solution ABBREVIATE In the equation for the Pythagorean Theorem, “length of hypotenuse” and “length of leg” was shortened to “hypotenuse” and “leg”. ✓ (hypotenuse)2 5 (leg)2 1 (leg)2 x Pythagorean Theorem x 2 5 62 1 8 2 Substitute. x 2 5 36 1 64 Multiply. 2 Add. x 5 10 Find the positive square root. x 5 100 GUIDED PRACTICE 6 for Example 1 Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle. Write your answer in simplest radical form. 1. x 3 5 2. x 4 6 7.1 Apply the Pythagorean Theorem 433 ★ EXAMPLE 2 Standardized Test Practice A 16 foot ladder rests against the side of the house, and the base of the ladder is 4 feet away. Approximately how high above the ground is the top of the ladder? A 240 feet B 20 feet C 16.5 feet D 15.5 feet FT XFT FT Solution 1ofLength ladder2 2 APPROXIMATE In real-world applications, it is usually appropriate to use a calculator to approximate the square root of a number. Round your answer to the nearest tenth. ✓ Distance 1from house2 2 5 1ofHeight ladder2 2 1 162 5 42 1 x2 Substitute. 256 5 16 1 x2 Multiply. 240 5 x 2 Subtract 16 from each side. } Ï240 5 x Find positive square root. 15.491 ø x Approximate with a calculator. The ladder is resting against the house at about 15.5 feet above the ground. c The correct answer is D. GUIDED PRACTICE A B C D for Example 2 3. The top of a ladder rests against a wall, 23 feet above the ground. The base of the ladder is 6 feet away from the wall. What is the length of the ladder? 4. The Pythagorean Theorem is only true for what type of triangle? PROVING THE PYTHAGOREAN THEOREM There are many proofs of the Pythagorean Theorem. An informal proof is shown below. You will write another proof in Exercise 32 on page 455. In the figure at the right, the four right triangles are congruent, and they form a small square in the middle. The area of the large square is equal to the area of the four triangles plus the area of the smaller square. REVIEW AREA Recall that the area of a square with side length s is A 5 s 2. The area of a triangle with base b and 1 height h is A 5 }bh. 2 Area of large square 1 (a 1 b)2 5 41 } ab 2 1 c 2 2 a 2 1 2ab 1 b 2 5 2ab 1 c 2 2 2 a 1b 5c (FPNFUSZ 434 Area of four triangles 5 2 at classzone.com Chapter 7 Right Triangles and Trigonometry 1 Area of smaller square Use area formulas. Multiply. Subtract 2ab from each side. a b b a c c c a b c b a EXAMPLE 3 Find the area of an isosceles triangle Find the area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters. Solution STEP 1 Draw a sketch. By definition, the length of an altitude is the height of a triangle. In an isosceles triangle, the altitude to the base is also a perpendicular bisector. So, the altitude divides the triangle into two right triangles with the dimensions shown. 13 m STEP 2 Use the Pythagorean Theorem to find the height h 5m 13 m 5m of the triangle. c 2 5 a2 1 b 2 2 2 13 5 5 1 h 2 169 5 25 1 h2 144 5 h READ TABLES You may find it helpful to use the Table of Squares and Square Roots on p. 924. 2 Pythagorean Theorem Substitute. Multiply. Subtract 25 from each side. 12 5 h Find the positive square root. STEP 3 Find the area. 1 1 Area 5 } (base)(height) 5 } (10)(12) 5 60 m 2 2 2 c The area of the triangle is 60 square meters. ✓ GUIDED PRACTICE for Example 3 Find the area of the triangle. 5. 18 ft 26 m 6. 30 ft 18 ft 20 m 26 m PYTHAGOREAN TRIPLES A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c 2 5 a2 1 b2. For Your Notebook KEY CONCEPT STANDARDIZED TESTS You may find it helpful to memorize the basic Pythagorean triples, shown in bold, for standardized tests. Common Pythagorean Triples and Some of Their Multiples 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 6, 8, 10 9, 12, 15 30, 40, 50 3x, 4x, 5x 10, 24, 26 15, 36, 39 50, 120, 130 5x, 12x, 13x 16, 30, 34 24, 45, 51 80, 150, 170 8x, 15x, 17x 14, 48, 50 21, 72, 75 70, 240, 250 7x, 24x, 25x The most common Pythagorean triples are in bold. The other triples are the result of multiplying each integer in a bold face triple by the same factor. 7.1 Apply the Pythagorean Theorem 435 EXAMPLE 4 Find the length of a hypotenuse using two methods Find the length of the hypotenuse of the right triangle. 24 Solution 10 x Method 1: Use a Pythagorean triple. A common Pythagorean triple is 5, 12, 13. Notice that if you multiply the lengths of the legs of the Pythagorean triple by 2, you get the lengths of the legs of this triangle: 5 p 2 5 10 and 12 p 2 5 24. So, the length of the hypotenuse is 13 p 2 5 26. Method 2: Use the Pythagorean Theorem. x2 5 102 1 242 2 x 5 100 1 576 Multiply. x2 5 676 Add. x 5 26 ✓ Pythagorean Theorem Find the positive square root. GUIDED PRACTICE for Example 4 Find the unknown side length of the right triangle using the Pythagorean Theorem. Then use a Pythagorean triple. 7. 9 in. 7.1 8. 12 in. x x 14 cm 48 cm EXERCISES HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 9, 11, and 33 ★ 5 STANDARDIZED TEST PRACTICE Exs. 2, 17, 27, 33, and 36 5 MULTIPLE REPRESENTATIONS Ex. 35 SKILL PRACTICE 1. VOCABULARY Copy and complete: A set of three positive integers a, b, and c that satisfy the equation c 2 5 a2 1 b2 is called a ? . 2. ALGEBRA Find the length of the hypotenuse of the right triangle. EXAMPLE 1 on p. 433 for Exs. 3–7 ★ WRITING Describe the information you need to have in order to use the Pythagorean Theorem to find the length of a side of a triangle. 3. 4. 50 x 120 436 Chapter 7 Right Triangles and Trigonometry 5. x 56 33 42 40 x ERROR ANALYSIS Describe and correct the error in using the Pythagorean Theorem. 6. 7. 26 x 7 10 24 24 x2 5 72 1 242 a 2 1 b2 5 c 2 x2 5 (7 1 24) 2 102 1 262 5 242 x2 5 312 x 5 31 EXAMPLE 2 on p. 434 for Exs. 8–10 FINDING A LENGTH Find the unknown leg length x. 8. 9. 10. 13.4 in. x 16.7 ft 9.8 in. x 5.7 ft 4.9 ft x 8.9 ft EXAMPLE 3 FINDING THE AREA Find the area of the isosceles triangle. on p. 435 for Exs. 11–13 11. 12. 17 m h 17 m 20 ft 20 ft h 13. 10 cm on p. 436 for Exs. 14–17 10 cm 32 ft 12 cm 16 m EXAMPLE 4 h FINDING SIDE LENGTHS Find the unknown side length of the right triangle using the Pythagorean Theorem or a Pythagorean triple. 14. x 17. 15. 72 21 16. 50 30 60 x x 68 ★ MULTIPLE CHOICE What is the length of the hypotenuse of a right triangle with leg lengths of 8 inches and 15 inches? A 13 inches B 17 inches C 21 inches D 25 inches PYTHAGOREAN TRIPLES The given lengths are two sides of a right triangle. All three side lengths of the triangle are integers and together form a Pythagorean triple. Find the length of the third side and tell whether it is a leg or the hypotenuse. 18. 24 and 51 19. 20 and 25 20. 28 and 96 21. 20 and 48 22. 75 and 85 23. 72 and 75 7.1 Apply the Pythagorean Theorem 437 7.4 Before Now Why? Key Vocabulary • isosceles triangle, Special Right Triangles You found side lengths using the Pythagorean Theorem. You will use the relationships among the sides in special right triangles. So you can find the height of a drawbridge, as in Ex. 28. A 458-458-908 triangle is an isosceles right triangle that can be formed by cutting a square in half as shown. p. 217 For Your Notebook THEOREM THEOREM 7.8 458-458-908 Triangle Theorem USE RATIOS The extended ratio of the side lengths of a 458-458-908 triangle is } 1: 1: Ï2. } In a 458-458-908 triangle, the hypotenuse is Ï 2 times as long as each leg. } hypotenuse 5 leg p Ï 2 x 2 458 x Proof: Ex. 30, p. 463 EXAMPLE 1 x 458 Find hypotenuse length in a 458-458-908 triangle Find the length of the hypotenuse. a. b. 8 3 2 3 2 458 Solution a. By the Triangle Sum Theorem, the measure of the third angle must be 458. Then the triangle is a 458-458-908 triangle, so by Theorem 7.8, the } hypotenuse is Ï2 times as long as each leg. } hypotenuse 5 leg p Ï2 } 5 8Ï2 REVIEW ALGEBRA Remember the following properties of radicals: Ïa p } } } Ïb 5 Ïa p b Ïa p a 5 a } For a review of radical expressions, see p. 874. 458-458-908 Triangle Theorem Substitute. b. By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a 458-458-908 triangle. } hypotenuse 5 leg p Ï2 } } 458-458-908 Triangle Theorem 5 3Ï2 p Ï 2 Substitute. 53p2 Product of square roots 56 Simplify. 7.4 Special Right Triangles 457 EXAMPLE 2 Find leg lengths in a 458-458-908 triangle Find the lengths of the legs in the triangle. 5 2 x x Solution By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a 458-458-908 triangle. } hypotenuse 5 leg p Ï 2 } } 5Ï 2 5 x p Ï 2 } Substitute. } 5Ï2 Ï2 xÏ 2 Ï2 } Divide each side by Ï 2 . } } 5 } } 55x ★ 45 8-45 8-90 8 Triangle Theorem EXAMPLE 3 Simplify. Standardized Test Practice Triangle WXY is a right triangle. Find the length of } WX . Y 25 cm 458 X ELIMINATE CHOICES You can eliminate choices C and D because the hypotenuse has to be longer than the leg. W } A 50 cm B 25Ï2 cm C 25 cm 25Ï2 D } cm } 2 Solution By the Corollary to the Triangle Sum Theorem, the triangle is a 458-458-908 triangle. } hypotenuse 5 leg p Ï2 } WX 5 25Ï 2 c The correct answer is B. ✓ GUIDED PRACTICE 458-458-908 Triangle Theorem Substitute. A B C D for Examples 1, 2, and 3 Find the value of the variable. 1. 2. 2 2 2 2 x y x 3. 8 8 d 8 8 4. Find the leg length of a 458-458-908 triangle with a hypotenuse length of 6. 458 Chapter 7 Right Triangles and Trigonometry A 308-608-908 triangle can be formed by dividing an equilateral triangle in half. For Your Notebook THEOREM THEOREM 7.9 308-608-908 Triangle Theorem USE RATIOS The extended ratio of the side lengths of a 308-608-908 triangle is } 1 : Ï3 : 2. In a 308-608-908 triangle, the hypotenuse is twice as long as the shorter } leg, and the longer leg is Ï 3 times as long as the shorter leg. hypotenuse 5 2 p shorter leg } longer leg 5 shorter leg p Ï3 x 608 2x 308 Proof: Ex. 32, p. 463 EXAMPLE 4 x 3 Find the height of an equilateral triangle LOGO The logo on the recycling bin at the right resembles an equilateral triangle with side lengths of 6 centimeters. What is the approximate height of the logo? Solution REVIEW MEDIAN Remember that in an equilateral triangle, the altitude to a side is also the median to that side. So, altitude } BD bisects } AC . Draw the equilateral triangle described. Its altitude forms the longer leg of two 308-608-908 triangles. The length h of the altitude is approximately the height of the logo. B } longer leg 5 shorter leg p Ï 3 6 cm h } h 5 3 p Ï3 ø 5.2 cm EXAMPLE 5 A 6 cm 608 608 C 3 cm D 3 cm Find lengths in a 308-608-908 triangle Find the values of x and y. Write your answer in simplest radical form. 608 y x 308 9 STEP 1 Find the value of x. } longer leg 5 shorter leg p Ï 3 } 9 5 xÏ 3 Substitute. 9 Ï3 } } } 5 x 9 Ï3 308-608-908 Triangle Theorem Divide each side by Ï 3 . } Multiply numerator and } denominator by Ï 3 . Ï3 Ï3 } } p } } 5 x } 9Ï 3 3 }5x Multiply fractions. } 3Ï 3 5 x Simplify. STEP 2 Find the value of y. hypotenuse 5 2 p shorter leg } } y 5 2 p 3Ï 3 5 6Ï 3 30 8-60 8-90 8 Triangle Theorem Substitute and simplify. 7.4 Special Right Triangles 459 EXAMPLE 6 Find a height DUMP TRUCK The body of a dump truck is raised to empty a load of sand. How high is the 14 foot body from the frame when it is tipped upward at the given angle? a. 458 angle b. 608 angle Solution a. When the body is raised 458 above the frame, the height h is the length of a leg of a 458-458-908 triangle. The length of the hypotenuse is 14 feet. } REWRITE MEASURES To write 9.9 ft in feet and inches, multiply the decimal part by 12. 12 p 0.9 5 10.8 So, 9.9 ft is about 9 feet 11 inches. 14 5 h p Ï 2 458-458-908 Triangle Theorem } } 5 h 14 Ï2 Divide each side by Ï 2 . 9.9 ø h Use a calculator to approximate. 458 } 14 ft c When the angle of elevation is 458, the body is about 9 feet 11 inches above the frame. b. When the body is raised 608, the height h is the length of the longer leg of a 308-608-908 triangle. The length of the hypotenuse is 14 feet. hypotenuse 5 2 p shorter leg 14 5 2 p s 308-608-908 Triangle Theorem 608 Substitute. 75s Divide each side by 2. } longer leg 5 shorter leg p Ï 3 } 308-608-908 Triangle Theorem h 5 7Ï 3 Substitute. h ø 12.1 Use a calculator to approximate. c When the angle of elevation is 608, the body is about 12 feet 1 inch above the frame. (FPNFUSZ ✓ GUIDED PRACTICE at classzone.com for Examples 4, 5, and 6 Find the value of the variable. 5. 6. 608 4 3 x 4 h 308 2 2 7. WHAT IF? In Example 6, what is the height of the body of the dump truck if it is raised 308 above the frame? 8. In a 308-608-908 triangle, describe the location of the shorter side. Describe the location of the longer side? 460 Chapter 7 Right Triangles and Trigonometry 14 ft 7.4 EXERCISES HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 5, 9, and 27 ★ 5 STANDARDIZED TEST PRACTICE Exs. 2, 6, 19, 22, 29, and 34 SKILL PRACTICE 1. VOCABULARY Copy and complete: A triangle with two congruent sides and a right angle is called ? . 2. EXAMPLES 1 and 2 on pp. 457–458 for Exs. 3–5 ★ WRITING Explain why the acute angles in an isosceles right triangle always measure 458. 45 8-45 8-90 8 TRIANGLES Find the value of x. Write your answer in simplest radical form. 3. 4. 7 458 EXAMPLE 3 6. on p. 458 for Exs. 6–7 ★ 5. 3 2 x 5 2 x x 5 2 x } MULTIPLE CHOICE Find the length of AC. } C } A 7Ï2 in. B 2Ï 7 in. } 7Ï 2 C } in. 458 } D Ï 14 in. 2 A B 7 in. 7. ISOSCELES RIGHT TRIANGLE The square tile shown has painted corners in the shape of congruent 458-458-908 triangles. What is the value of x? What is the side length of the tile? EXAMPLES 4 and 5 on p. 459 for Exs. 8–10 30 8-60 8-90 8 TRIANGLES Find the value of each variable. Write your answers in simplest radical form. 8. 9. 10. 3 3 12 3 308 y 9 x x 308 608 y y x SPECIAL RIGHT TRIANGLES Copy and complete the table. 11. 12. b 458 c d 608 458 a a b C 7 ? ? f e ? 11 ? ? ? 10 } Ï5 ? ? ? } 6Ï2 ? d e f 5 ? ? 308 ? ? 14 ? ? } 8Ï 3 ? ? ? 12 } 18Ï3 7.4 Special Right Triangles ? 461 ALGEBRA Find the value of each variable. Write your answers in simplest radical form. 13. 14. 15 608 24 y x 16. m n 17. r 18. t 458 608 18 (FPNFUSZ ★ 308 3 s 608 19. 15. 6 q 9 3 e 608 458 4 p f g u at classzone.com MULTIPLE CHOICE Which side lengths do not represent a 308-608-908 triangle? } 1 Ï3 A } , }, 1 2 } } } 5 5Ï 3 C } , }, 10 2 } B Ï 2 , Ï 6 , 2Ï 2 2 } D 3, 3Ï 3 , 6 2 ERROR ANALYSIS Describe and correct the error in finding the length of the hypotenuse. 20. 21. 7 3 7 5 45º 5 30º 22. ★ 5 2 WRITING Abigail solved Example 5 on page 459 in a different way. } } Instead of dividing each side by Ï 3 , she multiplied each side by Ï3 . Does her method work? Explain why or why not. ALGEBRA Find the value of each variable. Write your answers in simplest radical form. 23. 24. 308 g f 10 25. y x 8 1508 308 4 2 26. CHALLENGE n ABC is a 308-608-908 triangle. y Find the coordinates of A. A x 1 462 5 WORKED-OUT SOLUTIONS on p. WS1 y 1 C (23, 21) ★ 5 STANDARDIZED TEST PRACTICE B(3, 21) 608 x APPENDIX A REVIEW OF FUNDAMENTAL CONCEPTS OF ALGEBRA A.1 REAL NUMBERS AND THEIR PROPERTIES What you should learn • Represent and classify real numbers. • Order real numbers and use inequalities. • Find the absolute values of real numbers and find the distance between two real numbers. • Evaluate algebraic expressions. • Use the basic rules and properties of algebra. Real Numbers Real numbers are used in everyday life to describe quantities such as age, miles per gallon, and population. Real numbers are represented by symbols such as 4 3 !5, 9, 0, , 0.666 . . . , 28.21, "2, ", and " !32. 3 Here are some important subsets (each member of subset B is also a member of set A) of the real numbers. The three dots, called ellipsis points, indicate that the pattern continues indefinitely. !1, 2, 3, 4, . . .# Why you should learn it Real numbers are used to represent many real-life quantities. For example, in Exercises 83–88 on page A12, you will use real numbers to represent the federal deficit. Set of natural numbers !0, 1, 2, 3, 4, . . .# Set of whole numbers !. . . , !3, !2, !1, 0, 1, 2, 3, . . .# Set of integers A real number is rational if it can be written as the ratio p)q of two integers, where q $ 0. For instance, the numbers 1 1 125 # 0.3333 . . . # 0.3, # 0.125, and # 1.126126 . . . # 1.126 3 8 111 are rational. The decimal representation of a rational number either repeats (as in 173 1 55 # 3.145 ' or terminates (as in 2 # 0.5'. A real number that cannot be written as the ratio of two integers is called irrational. Irrational numbers have infinite nonrepeating decimal representations. For instance, the numbers "2 # 1.4142135 . . . & 1.41 " # 3.1415926 . . . & 3.14 and are irrational. (The symbol & means “is approximately equal to.”) Figure A.1 shows subsets of real numbers and their relationships to each other. Real numbers Classifying Real Numbers Determine which numbers in the set Irrational numbers Rational numbers Integers Negative integers Natural numbers FIGURE Example 1 $!13, ! Noninteger fractions (positive and negative) 1 3 5 8 % "5, !1, ! , 0, , "2, ", 7 are (a) natural numbers, (b) whole numbers, (c) integers, (d) rational numbers, and (e) irrational numbers. Solution a. Natural numbers: !7# b. Whole numbers: !0, 7# c. Integers: !!13, !1, 0, 7# Whole numbers Zero A.1 Subsets of real numbers $ % 1 5 d. Rational numbers: !13, !1, ! , 0, , 7 3 8 ! # e. Irrational numbers: ! "5, "2, " Now try Exercise 11. A1 A2 Appendix A Review of Fundamental Concepts of Algebra Real numbers are represented graphically on the real number line. When you draw a point on the real number line that corresponds to a real number, you are plotting the real number. The point 0 on the real number line is the origin. Numbers to the right of 0 are positive, and numbers to the left of 0 are negative, as shown in Figure A.2. The term nonnegative describes a number that is either positive or zero. Origin Negative direction FIGURE −4 −3 −2 −1 0 1 2 3 Positive direction 4 A.2 The real number line As illustrated in Figure A.3, there is a one-to-one correspondence between real numbers and points on the real number line. − 53 −3 −2 −1 0 −2.4 π 0.75 1 2 −3 3 Every real number corresponds to exactly one point on the real number line. FIGURE −2 2 −1 0 1 2 3 Every point on the real number line corresponds to exactly one real number. A.3 One-to-one correspondence Example 2 Plotting Points on the Real Number Line Plot the real numbers on the real number line. a. ! 7 4 b. 2.3 c. 2 3 d. !1.8 Solution All four points are shown in Figure A.4. − 1.8 − 74 −2 FIGURE 2 3 −1 0 2.3 1 2 3 A.4 a. The point representing the real number ! 74 # !1.75 lies between !2 and !1, but closer to !2, on the real number line. b. The point representing the real number 2.3 lies between 2 and 3, but closer to 2, on the real number line. c. The point representing the real number 23 # 0.666 . . . lies between 0 and 1, but closer to 1, on the real number line. d. The point representing the real number !1.8 lies between !2 and !1, but closer to !2, on the real number line. Note that the point representing !1.8 lies slightly to the left of the point representing ! 74. Now try Exercise 17. Appendix A.1 Real Numbers and Their Properties A3 Ordering Real Numbers One important property of real numbers is that they are ordered. Definition of Order on the Real Number Line a −1 If a and b are real numbers, a is less than b if b ! a is positive. The order of a and b is denoted by the inequality a < b. This relationship can also be described by saying that b is greater than a and writing b > a. The inequality a ≤ b means that a is less than or equal to b, and the inequality b ≥ a means that b is greater than or equal to a. The symbols <, >, %, and & are inequality symbols. b 0 1 2 Geometrically, this definition implies that a < b if and only if a lies to the left of b on the real number line, as shown in Figure A.5. FIGURE A.5 a < b if and only if a lies to the left of b. Example 3 −4 −3 FIGURE −4 −2 a. !3, 0 −2 −1 0 1 1 1 , 4 3 1 1 d. ! , ! 5 2 1 c. Because 41 lies to the left of 3 on the real number line, as shown in Figure A.8, you A.8 1 4 can say that is less than − 12 − 15 −1 FIGURE c. a. Because !3 lies to the left of 0 on the real number line, as shown in Figure A.6, you can say that !3 is less than 0, and write !3 < 0. b. Because !2 lies to the right of !4 on the real number line, as shown in Figure A.7, you can say that !2 is greater than !4, and write !2 > !4. 1 3 0 b. !2, !4 Solution A.7 1 4 FIGURE Place the appropriate inequality symbol (< or >' between the pair of real numbers. 0 A.6 −3 FIGURE −1 Ordering Real Numbers 1 3, and write 41 < 13. d. Because ! 15 lies to the right of ! 12 on the real number line, as shown in Figure A.9, you can say that ! 15 is greater than ! 12, and write ! 15 > ! 12. 0 A.9 Now try Exercise 25. Example 4 Interpreting Inequalities Describe the subset of real numbers represented by each inequality. a. x % 2 x≤2 x 0 FIGURE 1 2 3 4 A.10 −2 ≤ x < 3 x −2 −1 FIGURE A.11 0 1 2 3 b. !2 % x < 3 Solution a. The inequality x ≤ 2 denotes all real numbers less than or equal to 2, as shown in Figure A.10. b. The inequality !2 ≤ x < 3 means that x ≥ !2 and x < 3. This “double inequality” denotes all real numbers between !2 and 3, including !2 but not including 3, as shown in Figure A.11. Now try Exercise 31. A4 Appendix A Review of Fundamental Concepts of Algebra Inequalities can be used to describe subsets of real numbers called intervals. In the bounded intervals below, the real numbers a and b are the endpoints of each interval. The endpoints of a closed interval are included in the interval, whereas the endpoints of an open interval are not included in the interval. Bounded Intervals on the Real Number Line Notation +a, b* The reason that the four types of intervals at the right are called bounded is that each has a finite length. An interval that does not have a finite length is unbounded (see below). WARNING / CAUTION Whenever you write an interval containing ' or ! ', always use a parenthesis and never a bracket. This is because ' and ! ' are never an endpoint of an interval and therefore are not included in the interval. Interval Type Closed (a, b' Open +a, b' Inequality a % x % b Graph x a b a b a b a b a < x < b x a % x < b (a, b* x a < x % b x The symbols ', positive infinity, and ! ', negative infinity, do not represent real numbers. They are simply convenient symbols used to describe the unboundedness of an interval such as (1, '' or (! ', 3*. Unbounded Intervals on the Real Number Line Notation +a, '' Interval Type Inequality x & a Graph x a (a, '' Open x > a x a (! ', b* x % b x b (! ', b' Open (! ', '' Entire real line x < b x b Example 5 !' < x < ' Using Inequalities to Represent Intervals Use inequality notation to describe each of the following. a. c is at most 2. b. m is at least !3. c. All x in the interval (!3, 5* Solution a. The statement “c is at most 2” can be represented by c ≤ 2. b. The statement “m is at least !3” can be represented by m ≥ !3. c. “All x in the interval (!3, 5*” can be represented by !3 < x ≤ 5. Now try Exercise 45. x Appendix A.1 Example 6 Real Numbers and Their Properties A5 Interpreting Intervals Give a verbal description of each interval. a. (!1, 0' b. + 2, '' c. (! ', 0' Solution a. This interval consists of all real numbers that are greater than !1 and less than 0. b. This interval consists of all real numbers that are greater than or equal to 2. c. This interval consists of all negative real numbers. Now try Exercise 41. Absolute Value and Distance The absolute value of a real number is its magnitude, or the distance between the origin and the point representing the real number on the real number line. Definition of Absolute Value If a is a real number, then the absolute value of a is ,a, # $!a, a, if a ≥ 0 . if a < 0 Notice in this definition that the absolute value of a real number is never negative. For instance, if a # !5, then !5 # ! (!5' # 5. The absolute value of a real number is either positive or zero. Moreover, 0 is the only real number whose absolute value is 0. So, 0 # 0. , , ,, Example 7 Finding Absolute Values , , ,!4.3, # 4.3 ,, 2 2 # 3 3 a. !15 # 15 b. c. d. ! !6 # ! (6' # !6 , , Now try Exercise 51. Example 8 Evaluate Evaluating the Absolute Value of a Number ,x, for (a) x > 0 and (b) x < 0. x Solution ,, a. If x > 0, then x # x and ,, ,x, # x # 1. x b. If x < 0, then x # !x and x , , # !x # !1. x x Now try Exercise 59. x A6 Appendix A Review of Fundamental Concepts of Algebra The Law of Trichotomy states that for any two real numbers a and b, precisely one of three relationships is possible: a # b, a < b, Example 9 or a > b. Law of Trichotomy Comparing Real Numbers Place the appropriate symbol (<, >, or =) between the pair of real numbers. , ,!,3, , a. !4 ,!,10, , ,!,!7, b. !10 Solution , , ,, , , , , , , , , c. ! !7 , , ,, , , , , , , , , a. !4 > 3 because !4 # 4 and 3 # 3, and 4 is greater than 3. b. !10 # 10 because !10 # 10 and 10 # 10. c. ! !7 < !7 because ! !7 # !7 and !7 # 7, and !7 is less than 7. Now try Exercise 61. Properties of Absolute Values ,, 2. !a # a , , , ,, , 4. 1. a & 0 3. ab # a b −2 −1 0 ,, Absolute value can be used to define the distance between two points on the real number line. For instance, the distance between !3 and 4 is 7 −3 , , ,, a ,a,, b $ 0 # b ,b, 1 2 3 4 A.12 The distance between !3 and 4 is 7. ,!3 ! 4, # ,!7, #7 FIGURE as shown in Figure A.12. Distance Between Two Points on the Real Number Line Let a and b be real numbers. The distance between a and b is , , , , d(a, b' # b ! a # a ! b . Example 10 Finding a Distance Find the distance between !25 and 13. Solution The distance between !25 and 13 is given by ,!25 ! 13, # ,!38, # 38. Distance between !25 and 13 The distance can also be found as follows. ,13 ! (!25', # ,38, # 38 Now try Exercise 67. Distance between !25 and 13 Appendix A.1 Real Numbers and Their Properties A7 Algebraic Expressions One characteristic of algebra is the use of letters to represent numbers. The letters are variables, and combinations of letters and numbers are algebraic expressions. Here are a few examples of algebraic expressions. 5x, 2x ! 3, x2 4 , (2 7x ( y Definition of an Algebraic Expression An algebraic expression is a collection of letters (variables) and real numbers (constants) combined using the operations of addition, subtraction, multiplication, division, and exponentiation. The terms of an algebraic expression are those parts that are separated by addition. For example, x 2 ! 5x ( 8 # x 2 ( (!5x' ( 8 has three terms: x 2 and !5x are the variable terms and 8 is the constant term. The numerical factor of a term is called the coefficient. For instance, the coefficient of !5x is !5, and the coefficient of x 2 is 1. Example 11 Identifying Terms and Coefficients Algebraic Expression 1 7 b. 2x2 ! 6x ( 9 3 1 c. ( x4 ! y x 2 a. 5x ! Terms Coefficients 1 7 2x2, !6x, 9 3 1 4 , x , !y x 2 1 7 2, !6, 9 1 3, , !1 2 5x, ! 5, ! Now try Exercise 89. To evaluate an algebraic expression, substitute numerical values for each of the variables in the expression, as shown in the next example. Example 12 Evaluating Algebraic Expressions Expression a. !3x ( 5 b. 3x 2 ( 2x ! 1 2x c. x(1 Value of Variable x#3 x # !1 x # !3 Substitute Value of Expression !3(3' ( 5 3(!1'2 ( 2(!1' ! 1 2(!3' !3 ( 1 !9 ( 5 # !4 3!2!1#0 !6 #3 !2 Note that you must substitute the value for each occurrence of the variable. Now try Exercise 95. When an algebraic expression is evaluated, the Substitution Principle is used. It states that “If a # b, then a can be replaced by b in any expression involving a.” In Example 12(a), for instance, 3 is substituted for x in the expression !3x ( 5. A8 Appendix A Review of Fundamental Concepts of Algebra Basic Rules of Algebra There are four arithmetic operations with real numbers: addition, multiplication, subtraction, and division, denoted by the symbols (, + or * , !, and ) or /. Of these, addition and multiplication are the two primary operations. Subtraction and division are the inverse operations of addition and multiplication, respectively. Definitions of Subtraction and Division Subtraction: Add the opposite. a ! b # a ( (!b' Division: Multiply by the reciprocal. If b $ 0, then a)b # a -b. # b . 1 a In these definitions, !b is the additive inverse (or opposite) of b, and 1)b is the multiplicative inverse (or reciprocal) of b. In the fractional form a)b, a is the numerator of the fraction and b is the denominator. Because the properties of real numbers below are true for variables and algebraic expressions as well as for real numbers, they are often called the Basic Rules of Algebra. Try to formulate a verbal description of each property. For instance, the first property states that the order in which two real numbers are added does not affect their sum. Basic Rules of Algebra Let a, b, and c be real numbers, variables, or algebraic expressions. Property Commutative Property of Addition: Commutative Property of Multiplication: Associative Property of Addition: Associative Property of Multiplication: Distributive Properties: Additive Identity Property: Multiplicative Identity Property: Additive Inverse Property: Multiplicative Inverse Property: Example a(b#b(a ab # ba (a ( b' ( c # a ( (b ( c' (ab' c # a(bc' a(b ( c' # ab ( ac (a ( b'c # ac ( bc a(0#a a*1#a a ( (!a' # 0 1 a * # 1, a $ 0 a x2 4x ( # x 2 ( 4x (4 ! x' x 2 # x 2(4 ! x' (x ( 5' ( x 2 # x ( (5 ( x 2' (2x * 3y'(8' # (2x'(3y * 8' 3x(5 ( 2x' # 3x * 5 ( 3x * 2x ( y ( 8' y # y * y ( 8 * y 5y 2 ( 0 # 5y 2 (4x 2'(1' # 4x 2 5x 3 ( (!5x 3' # 0 1 (x 2 ( 4' 2 #1 x (4 - . Because subtraction is defined as “adding the opposite,” the Distributive Properties are also true for subtraction. For instance, the “subtraction form” of a(b ( c' # ab ( ac is a(b ! c' # ab ! ac. Note that the operations of subtraction and division are neither commutative nor associative. The examples 7 ! 3 $ 3 ! 7 and 20 ) 4 $ 4 ) 20 show that subtraction and division are not commutative. Similarly 5 ! (3 ! 2' $ (5 ! 3' ! 2 and 16 ) (4 ) 2) $ (16 ) 4) ) 2 demonstrate that subtraction and division are not associative. Appendix A.1 Example 13 Real Numbers and Their Properties A9 Identifying Rules of Algebra Identify the rule of algebra illustrated by the statement. a. (5x3'2 # 2(5x3' b. -4x ( 31. ! -4x ( 31. # 0 1 # 1, x $ 0 7x d. (2 ( 5x2' ( x2 # 2 ( (5x2 ( x2' c. 7x * Solution a. This statement illustrates the Commutative Property of Multiplication. In other words, you obtain the same result whether you multiply 5x3 by 2, or 2 by 5x3. b. This statement illustrates the Additive Inverse Property. In terms of subtraction, this property simply states that when any expression is subtracted from itself the result is 0. c. This statement illustrates the Multiplicative Inverse Property. Note that it is important that x be a nonzero number. If x were 0, the reciprocal of x would be undefined. d. This statement illustrates the Associative Property of Addition. In other words, to form the sum 2 ( 5x2 ( x2 it does not matter whether 2 and 5x2, or 5x2 and x2 are added first. Now try Exercise 101. Properties of Negation and Equality Let a, b, and c be real numbers, variables, or algebraic expressions. Notice the difference between the opposite of a number and a negative number. If a is already negative, then its opposite, !a, is positive. For instance, if a # !5, then !a # !(!5) # 5. Property 1. (!1' a # !a Example (!1'7 # !7 2. ! (!a' # a ! (!6' # 6 3. (!a'b # ! (ab' # a(!b' (!5'3 # ! (5 * 3' # 5(!3' 4. (!a'(!b' # ab (!2'(!x' # 2x 5. ! (a ( b' # (!a' ( (!b' ! (x ( 8' # (!x' ( (!8' 6. If a # b, then a ± c # b ± c. 1 2 7. If a # b, then ac # bc. 42 8. If a ± c # b ± c, then a # b. 1.4 ! 1 # 75 ! 1 ⇒ 1.4 # 75 9. If ac # bc and c $ 0, then a # b. 3x # 3 # !x ! 8 ( 3 # 0.5 ( 3 * 2 # 16 * 2 *4 ⇒ x#4 A10 Appendix A Review of Fundamental Concepts of Algebra Properties of Zero The “or” in the Zero-Factor Property includes the possibility that either or both factors may be zero. This is an inclusive or, and it is the way the word “or” is generally used in mathematics. Let a and b be real numbers, variables, or algebraic expressions. 1. a ( 0 # a and a ! 0 # a 3. 0 # 0, a 2. a 4. a$0 *0#0 a is undefined. 0 5. Zero-Factor Property: If ab # 0, then a # 0 or b # 0. Properties and Operations of Fractions Let a, b, c, and d be real numbers, variables, or algebraic expressions such that b $ 0 and d $ 0. 1. Equivalent Fractions: 2. Rules of Signs: ! a c # if and only if ad # bc. b d a !a a !a a # # and # b b !b !b b 3. Generate Equivalent Fractions: a ac # , c$0 b bc 4. Add or Subtract with Like Denominators: a c a±c ± # b b b 5. Add or Subtract with Unlike Denominators: In Property 1 of fractions, the phrase “if and only if” implies two statements. One statement is: If a)b # c)d, then ad # bc. The other statement is: If ad # bc, where b $ 0 and d $ 0, then a)b # c)d. 6. Multiply Fractions: 7. Divide Fractions: Example 14 a b c a c ad ± bc ± # b d bd ac * d # bd a c a ) # b d b d ad * c # bc , c$0 Properties and Operations of Fractions a. Equivalent fractions: x 3 * x 3x # # 5 3 * 5 15 c. Add fractions with unlike denominators: b. Divide fractions: 7 3 7 2 14 ) # * # x 2 x 3 3x x 2x 5 * x ( 3 * 2x 11x ( # # 3 5 3*5 15 Now try Exercise 119. If a, b, and c are integers such that ab # c, then a and b are factors or divisors of c. A prime number is an integer that has exactly two positive factors—itself and 1—such as 2, 3, 5, 7, and 11. The numbers 4, 6, 8, 9, and 10 are composite because each can be written as the product of two or more prime numbers. The number 1 is neither prime nor composite. The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be written as the product of prime numbers in precisely one way (disregarding order). For instance, the prime factorization of 24 is 24 # 2 * 2 * 2 * 3. Appendix A.1 A.1 EXERCISES Real Numbers and Their Properties A11 See www.CalcChat.com for worked-out solutions to odd-numbered exercises. VOCABULARY: Fill in the blanks. p of two integers, where q $ 0. q ________ numbers have infinite nonrepeating decimal representations. The point 0 on the real number line is called the ________. The distance between the origin and a point representing a real number on the real number line is the ________ ________ of the real number. A number that can be written as the product of two or more prime numbers is called a ________ number. An integer that has exactly two positive factors, the integer itself and 1, is called a ________ number. An algebraic expression is a collection of letters called ________ and real numbers called ________. The ________ of an algebraic expression are those parts separated by addition. The numerical factor of a variable term is the ________ of the variable term. The ________ ________ states that if ab # 0, then a # 0 or b # 0. 1. A real number is ________ if it can be written as the ratio 2. 3. 4. 5. 6. 7. 8. 9. 10. SKILLS AND APPLICATIONS In Exercises 11–16, determine which numbers in the set are (a) natural numbers, (b) whole numbers, (c) integers, (d) rational numbers, and (e) irrational numbers. In Exercises 25–30, plot the two real numbers on the real number line. Then place the appropriate inequality symbol (< or >) between them. 11. 12. 13. 14. 15. 16. 25. !4, !8 27. 32, 7 26. !3.5, 1 16 28. 1, 3 2 29. 56, 3 8 3 30. ! 7, ! 7 !!9, ! 72, 5, 23, "2, 0, 1, !4, 2, !11# !"5, !7, ! 73, 0, 3.12, 54 , !3, 12, 5# !2.01, 0.666 . . . , !13, 0.010110111 . . . , 1, !6# !2.3030030003 . . . , 0.7575, !4.63, "10, !75, 4# !! ", ! 13, 63, 12"2, !7.5, !1, 8, !22# !25, !17, ! 125, "9, 3.12, 12", 7, !11.1, 13# In Exercises 17 and 18, plot the real numbers on the real number line. 17. (a) 3 (b) 72 18. (a) 8.5 (b) 5 (c) ! 2 (d) !5.2 4 8 3 (c) !4.75 (d) ! 3 In Exercises 19–22, use a calculator to find the decimal form of the rational number. If it is a nonterminating decimal, write the repeating pattern. 19. 21. 5 8 41 333 20. 22. 1 3 6 11 24. −3 −7 −2 −6 −1 −5 0 −4 1 −3 −2 31. 33. 35. 37. 39. 41. x % 5 x < 0 +4, '' !2 < x < 2 !1 ≤ x < 0 +!2, 5' 32. 34. 36. 38. 40. 42. x & !2 x > 3 (! ', 2' 0 ≤ x ≤ 5 0 < x ≤ 6 (!1, 2* In Exercises 43–50, use inequality notation and interval notation to describe the set. In Exercises 23 and 24, approximate the numbers and place the correct symbol (< or >) between them. 23. In Exercises 31– 42, (a) give a verbal description of the subset of real numbers represented by the inequality or the interval, (b) sketch the subset on the real number line, and (c) state whether the interval is bounded or unbounded. 2 3 −1 0 43. 44. 45. 46. 47. 48. 49. 50. y is nonnegative. y is no more than 25. x is greater than !2 and at most 4. y is at least !6 and less than 0. t is at least 10 and at most 22. k is less than 5 but no less than !3. The dog’s weight W is more than 65 pounds. The annual rate of inflation r is expected to be at least 2.5% but no more than 5%. Appendix A Review of Fundamental Concepts of Algebra In Exercises 51–60, evaluate the expression. 51. 52. 53. 54. 55. 56. ,!10, ,0, ,3 ! 8, ,4 ! 1, ,!1, ! ,!2, !3 ! ,!3, !5 !5 58. !3 !3 57. 59. , , , , x ( , 2,, x < !2 x(2 x!1 60. , x > 1 x!1 , , In Exercises 61–66, place the correct symbol (<, >, or #' between the two real numbers. 61. 62. 63. 64. 65. 66. ,!3,!! ,!3, ,!4,!,4, !5!! ,5, ! ,!6,!,!6, ! ,!2,!! ,2, BUDGET VARIANCE In Exercises 79–82, the accounting department of a sports drink bottling company is checking to see whether the actual expenses of a department differ from the budgeted expenses by more than $500 or by more than 5%. Fill in the missing parts of the table, and determine whether each actual expense passes the “budget variance test.” 79. 80. 81. 82. In Exercises 73–78, use absolute value notation to describe the situation. 73. 74. 75. 76. 77. The distance between x and 5 is no more than 3. The distance between x and !10 is at least 6. y is at least six units from 0. y is at most two units from a. While traveling on the Pennsylvania Turnpike, you pass milepost 57 near Pittsburgh, then milepost 236 near Gettysburg. How many miles do you travel during that time period? 78. The temperature in Bismarck, North Dakota was 60,F at noon, then 23,F at midnight. What was the change in temperature over the 12-hour period? ,a ! b, ! ! ! ! 2600 !(!2)!!2 a # 126, b # 75 a # !126, b # !75 a # ! 52, b # 0 a # 14, b # 11 4 112 a # 16 5 , b # 75 a # 9.34, b # !5.65 Budgeted Actual Expense, b Expense, a $112,700 $113,356 $9,400 $9,772 $37,640 $37,335 $2,575 $2,613 0.05b ! ! ! ! FEDERAL DEFICIT In Exercises 83–88, use the bar graph, which shows the receipts of the federal government (in billions of dollars) for selected years from 1996 through 2006. In each exercise you are given the expenditures of the federal government. Find the magnitude of the surplus or deficit for the year. (Source: U.S. Office of Management and Budget) In Exercises 67–72, find the distance between a and b. 67. 68. 69. 70. 71. 72. Wages Utilities Taxes Insurance Receipts (in billions of dollars) A12 2407.3 2400 2200 2025.5 2000 1800 1600 1400 1722.0 1853.4 1880.3 1453.2 1200 1996 1998 2000 2002 2004 2006 Year Year Receipts 83. 84. 85. 86. 87. 88. 1996 1998 2000 2002 2004 2006 ! ! ! ! ! ! Expenditures |Receipts ! Expenditures| $1560.6 billion $1652.7 billion $1789.2 billion $2011.2 billion $2293.0 billion $2655.4 billion ! ! ! ! ! ! In Exercises 89–94, identify the terms. Then identify the coefficients of the variable terms of the expression. 89. 7x ( 4 91. "3x 2 ! 8x ! 11 x 93. 4x 3 ( ! 5 2 90. 6x 3 ! 5x 92. 3"3x 2 ( 1 x2 94. 3x 4 ! 4 Appendix A.1 In Exercises 95–100, evaluate the expression for each value of x. (If not possible, state the reason.) 95. 96. 97. 98. 99. 100. Expression 4x ! 6 9 ! 7x x 2 ! 3x ( 4 !x 2 ( 5x ! 4 x(1 x!1 x x(2 (a) (a) (a) (a) Values x # !1 (b) x # !3 (b) x # !2 (b) x # !1 (b) x#0 x#3 x#2 x#1 (a) x # 1 (b) x # !1 (a) x # 2 (b) x # !2 In Exercises 101–112, identify the rule(s) of algebra illustrated by the statement. 101. x ( 9 # 9 ( x 102. 2( 12 ' # 1 1 103. (h ( 6' # 1, h $ !6 h(6 104. (x ( 3' ! (x ( 3' # 0 105. 2(x ( 3' # 2 * x ( 2 * 3 106. (z ! 2' ( 0 # z ! 2 107. 1 * (1 ( x' # 1 ( x 108. (z ( 5'x # z * x ( 5 * x 109. x ( ( y ( 10' # (x ( y' ( 10 110. x(3y' # (x * 3'y # (3x' y 111. 3(t ! 4' # 3 * t ! 3 * 4 1 112. 7(7 * 12' # ( 17 * 7'12 # 1 * 12 # 12 In Exercises 113–120, perform the operation(s). (Write fractional answers in simplest form.) 5 3 113. 16 ( 16 5 115. 58 ! 12 ( 16 1 117. 12 ) 4 119. 114. 76 ! 47 6 13 116. 10 11 ( 33 ! 66 4 118. ! (6 * 8 ' 2x x ! 3 4 120. 5x 6 2 *9 EXPLORATION In Exercises 121 and 122, use the real numbers A, B, and C shown on the number line. Determine the sign of each expression. C B A 0 121. (a) !A (b) B ! A 122. (a) !C (b) A ! C A13 Real Numbers and Their Properties 123. CONJECTURE (a) Use a calculator to complete the table. 1 n 0.5 0.01 0.0001 0.000001 5)n (b) Use the result from part (a) to make a conjecture about the value of 5)n as n approaches 0. 124. CONJECTURE (a) Use a calculator to complete the table. 1 n 10 100 10,000 100,000 5)n (b) Use the result from part (a) to make a conjecture about the value of 5)n as n increases without bound. TRUE OR FALSE? In Exercises 125–128, determine whether the statement is true or false. Justify your answer. 125. If a > 0 and b < 0, then a ! b > 0. 126. If a > 0 and b < 0, then ab > 0. 127. If a < b, then 128. Because 1 1 < , where a $ 0 and b $ 0. a b a(b a b c c c # ( , then # ( . c c c a(b a b , , ,, ,, 129. THINK ABOUT IT Consider u ( v and u ( v , where u $ 0 and v $ 0. (a) Are the values of the expressions always equal? If not, under what conditions are they unequal? (b) If the two expressions are not equal for certain values of u and v, is one of the expressions always greater than the other? Explain. 130. THINK ABOUT IT Is there a difference between saying that a real number is positive and saying that a real number is nonnegative? Explain. 131. THINK ABOUT IT Because every even number is divisible by 2, is it possible that there exist any even prime numbers? Explain. 132. THINK ABOUT IT Is it possible for a real number to be both rational and irrational? Explain. 133. WRITING Can it ever be true that a # !a for a real number a? Explain. ,, 134. CAPSTONE Describe the differences among the sets of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. A14 Appendix A Review of Fundamental Concepts of Algebra A.2 What you should learn • Use properties of exponents. • Use scientific notation to represent real numbers. • Use properties of radicals. • Simplify and combine radicals. • Rationalize denominators and numerators. • Use properties of rational exponents. EXPONENTS AND RADICALS Integer Exponents Repeated multiplication can be written in exponential form. Repeated Multiplication a*a*a*a*a Exponential Form a5 (!4'(!4'(!4' (!4'3 (2x'(2x'(2x'(2x' (2x'4 Why you should learn it Exponential Notation Real numbers and algebraic expressions are often written with exponents and radicals. For instance, in Exercise 121 on page A26, you will use an expression involving rational exponents to find the times required for a funnel to empty for different water heights. If a is a real number and n is a positive integer, then an # a * a *a. . .a n factors where n is the exponent and a is the base. The expression an is read “a to the nth power.” An exponent can also be negative. In Property 3 below, be sure you see how to use a negative exponent. Properties of Exponents T E C H N O LO G Y You can use a calculator to evaluate exponential expressions. When doing so, it is important to know when to use parentheses because the calculator follows the order of operations. For instance, evaluate (!2'4 as follows. Scientific: / 2 yx 0 #$! 4 " / /!0 2 0 > Graphing: 4 Let a and b be real numbers, variables, or algebraic expressions, and let m and n be integers. (All denominators and bases are nonzero.) Example # 32(4 # 36 # 729 1. 32 2. am # am!n an x7 # x7! 4 # x 3 x4 3. a!n # -. 1 1 # an a 4. a0 # 1, * 34 n a$0 y!4 # -. 1 1 # y4 y 4 (x 2 ( 1'0 # 1 5. (ab'm # am bm (5x'3 # 53x3 # 125x3 6. (am'n # amn ( y3'!4 # y3(!4) # y!12 # ENTER The display will be 16. If you omit the parentheses, the display will be !16. Property # a m(n a ma n 7. -b. a m # am bm , , ,, 8. a2 # a 2 # a2 -x. 2 3 # 1 y12 23 8 # x3 x3 ,(!2'2, # ,!2,2 # (2'2 # 4 Appendix A.2 Exponents and Radicals A15 It is important to recognize the difference between expressions such as (!2'4 and In (!2'4, the parentheses indicate that the exponent applies to the negative sign as well as to the 2, but in !24 # ! (24', the exponent applies only to the 2. So, (!2'4 # 16 and !24 # !16. The properties of exponents listed on the preceding page apply to all integers m and n, not just to positive integers, as shown in the examples in this section. !24. Example 1 Evaluating Exponential Expressions a. (!5'2 # (!5'(!5' # 25 Negative sign is part of the base. b. !52 # ! (5'(5' # !25 Negative sign is not part of the base. c. 2 Property 1 d. * 2 4 # 21(4 # 25 # 32 4 4 1 1 # 44!6 # 4!2 # 2 # 46 4 16 Properties 2 and 3 Now try Exercise 11. Example 2 Evaluating Algebraic Expressions Evaluate each algebraic expression when x # 3. a. 5x!2 b. 1 (!x'3 3 Solution a. When x # 3, the expression 5x!2 has a value of 5x!2 # 5(3'!2 # 5 5 # . 2 3 9 1 b. When x # 3, the expression (!x'3 has a value of 3 1 1 1 (!x'3 # (!3'3 # (!27' # !9. 3 3 3 Now try Exercise 23. Example 3 Using Properties of Exponents Use the properties of exponents to simplify each expression. a. (!3ab4'(4ab!3' b. (2xy2'3 c. 3a(!4a2'0 Solution a. (!3ab4'(4ab!3' # (!3'(4'(a'(a'(b4'(b!3' # !12a 2b b. (2xy 2'3 # 23(x'3( y 2'3 # 8x3y6 c. 3a(!4a 2'0 # 3a(1' # 3a, a $ 0 5x 3 2 52(x 3'2 25x 6 d. -y. # y2 # y2 Now try Exercise 31. d. -5xy . 3 2 A16 Appendix A Review of Fundamental Concepts of Algebra Example 4 Rarely in algebra is there only one way to solve a problem. Don’t be concerned if the steps you use to solve a problem are not exactly the same as the steps presented in this text. The important thing is to use steps that you understand and, of course, steps that are justified by the rules of algebra. For instance, you might prefer the following steps for Example 4(d). - . 3x 2 y !2 - . y # 3x 2 2 y2 # 4 9x Note how Property 3 is used in the first step of this solution. The fractional form of this property is -. a b !m Rewrite each expression with positive exponents. a. x!1 b. 1 3x!2 c. 12a3b!4 4a!2b d. -3xy . 2 !2 Solution 1 x a. x!1 # Property 3 b. 1 1(x 2' x 2 # # 3x!2 3 3 The exponent !2 does not apply to 3. c. 12a3b!4 12a3 * a2 # 4a!2b 4b * b4 Property 3 d. -3xy . # 3a5 b5 Property 1 # 3!2(x 2'!2 y!2 Properties 5 and 7 # 3!2x!4 y!2 Property 6 # y2 32x 4 Property 3 # y2 9x 4 Simplify. 2 !2 -.. b # a Rewriting with Positive Exponents m Now try Exercise 41. Scientific Notation Exponents provide an efficient way of writing and computing with very large (or very small) numbers. For instance, there are about 359 billion billion gallons of water on Earth—that is, 359 followed by 18 zeros. 359,000,000,000,000,000,000 It is convenient to write such numbers in scientific notation. This notation has the form ± c + 10n, where 1 ≤ c < 10 and n is an integer. So, the number of gallons of water on Earth can be written in scientific notation as 3.59 + 100,000,000,000,000,000,000 # 3.59 + 1020. The positive exponent 20 indicates that the number is large (10 or more) and that the decimal point has been moved 20 places. A negative exponent indicates that the number is small (less than 1). For instance, the mass (in grams) of one electron is approximately 9.0 + 10!28 # 0.0000000000000000000000000009. 28 decimal places Appendix A.2 Example 5 Exponents and Radicals A17 Scientific Notation Write each number in scientific notation. a. 0.0000782 b. 836,100,000 Solution a. 0.0000782 # 7.82 + 10!5 b. 836,100,000 # 8.361 + 108 Now try Exercise 45. Example 6 Decimal Notation Write each number in decimal notation. a. !9.36 + 10!6 b. 1.345 + 102 Solution a. !9.36 + 10!6 # !0.00000936 b. 1.345 + 102 # 134.5 Now try Exercise 55. T E C H N O LO G Y Most calculators automatically switch to scientific notation when they are showing large (or small) numbers that exceed the display range. To enter numbers in scientific notation, your calculator should have an exponential entry key labeled EE or EXP . Consult the user’s guide for your calculator for instructions on keystrokes and how numbers in scientific notation are displayed. Example 7 Evaluate Using Scientific Notation (2,400,000,000'(0.0000045' . (0.00003'(1500' Solution Begin by rewriting each number in scientific notation and simplifying. (2,400,000,000'(0.0000045' (2.4 + 109'(4.5 + 10!6' # (0.00003'(1500' (3.0 + 10!5'(1.5 + 103' # (2.4'(4.5'(103' (4.5'(10!2' # (2.4'(105' # 240,000 Now try Exercise 63(b). A18 Appendix A Review of Fundamental Concepts of Algebra Radicals and Their Properties A square root of a number is one of its two equal factors. For example, 5 is a square root of 25 because 5 is one of the two equal factors of 25. In a similar way, a cube root of a number is one of its three equal factors, as in 125 # 53. Definition of nth Root of a Number Let a and b be real numbers and let n & 2 be a positive integer. If a # bn then b is an nth root of a. If n # 2, the root is a square root. If n # 3, the root is a cube root. Some numbers have more than one nth root. For example, both 5 and !5 are square roots of 25. The principal square root of 25, written as "25, is the positive root, 5. The principal nth root of a number is defined as follows. Principal nth Root of a Number Let a be a real number that has at least one nth root. The principal nth root of a is the nth root that has the same sign as a. It is denoted by a radical symbol n a. " Principal nth root The positive integer n is the index of the radical, and the number a is the radicand. 2 a. (The plural of index is If n # 2, omit the index and write "a rather than " indices.) A common misunderstanding is that the square root sign implies both negative and positive roots. This is not correct. The square root sign implies only a positive root. When a negative root is needed, you must use the negative sign with the square root sign. Incorrect: "4 # ± 2 Example 8 Correct: ! "4 # !2 and "4 # 2 Evaluating Expressions Involving Radicals a. "36 # 6 because 62 # 36. b. ! "36 # !6 because ! ("36' # ! ("62' # ! (6' # !6. 5 3 53 125 125 5 c. 3 # because # 3# . 64 4 4 4 64 5 !32 # !2 because (!2'5 # !32. d. " 4 !81 is not a real number because there is no real number that can be raised to the e. " fourth power to produce !81. " -. Now try Exercise 65. Appendix A.2 Exponents and Radicals A19 Here are some generalizations about the nth roots of real numbers. Generalizations About nth Roots of Real Numbers Real Number a Integer n Root(s) of a Example a > 0 n > 0, n is even. n a, n a " !" 4 81 # 3, 4 81 # !3 " !" a > 0 or a < 0 n is odd. n " a 3 " !8 # !2 a < 0 n is even. No real roots "!4 is not a real number. a#0 n is even or odd. n 0 # 0 " 5 0 # 0 " Integers such as 1, 4, 9, 16, 25, and 36 are called perfect squares because they have integer square roots. Similarly, integers such as 1, 8, 27, 64, and 125 are called perfect cubes because they have integer cube roots. Properties of Radicals Let a and b be real numbers, variables, or algebraic expressions such that the indicated roots are real numbers, and let m and n be positive integers. 1. Property m n a # (" a' n 2. " a 3. 4. n " a n " b n n b#" ab *" # " n a , b Example 2 8 ' # (2'2 # 4 8 #( 3 2 " n m " "5 * "7 # "5 * 7 # "35 4 " 27 b$0 4 " 9 m n "a # mn "a " 3 " # "279 # 4 4 " 3 3 6 "10 # " " 10 n 5. (" a' # a ("3 '2 # 3 n ,, , , "(!12'2 # !12 # 12 n n 6. For n even, " a # a. 3 " (!12'3 # !12 n n For n odd, " a # a. ,, A common special case of Property 6 is "a2 # a . Example 9 Using Properties of Radicals Use the properties of radicals to simplify each expression. a. "8 * "2 3 5 b. (" ' 3 Solution a. b. c. d. "8 ( * "2 # "8 * 2 # "16 # 4 3 5 " '3 # 5 3 x3 # x " ,, 6 y6 # y " Now try Exercise 77. 3 x3 c. " 6 y6 d. " A20 Appendix A Review of Fundamental Concepts of Algebra Simplifying Radicals An expression involving radicals is in simplest form when the following conditions are satisfied. 1. All possible factors have been removed from the radical. 2. All fractions have radical-free denominators (accomplished by a process called rationalizing the denominator). 3. The index of the radical is reduced. To simplify a radical, factor the radicand into factors whose exponents are multiples of the index. The roots of these factors are written outside the radical, and the “leftover” factors make up the new radicand. WARNING / CAUTION Example 10 When you simplify a radical, it is important that both expressions are defined for the same values of the variable. For instance, in Example 10(b), "75x3 and 5x"3x are both defined only for nonnegative values of x. Similarly, in Example 10(c), 4 (5x'4 and 5 x are both " defined for all real values of x. Perfect 4th power ,, Simplifying Even Roots Leftover factor 4 4 a. " 48 # " 16 4 4 4 2 * 3 # 2" 3 *3#" Perfect square Leftover factor b. "75x3 # "25x 2 * 3x # "(5x'2 * 3x # 5x"3x 4 4 c. "(5x' # 5x # 5 x , , Find largest square factor. Find root of perfect square. ,, Now try Exercise 79(a). Example 11 Simplifying Odd Roots Perfect cube Leftover factor 3 3 a. " 24 # " 8 3 3 3 2 * 3 # 2" 3 *3#" Perfect cube Leftover factor 3 3 b. " 24a4 # " 8a3 * 3a 3 (2a'3 #" * 3a 3 3a # 2a " 3 !40x6 # " 3 (!8x6' c. " *5 3 (!2x 2'3 #" 3 5 # !2x 2 " Find largest cube factor. Find root of perfect cube. Find largest cube factor. *5 Find root of perfect cube. Now try Exercise 79(b). Appendix A.2 Exponents and Radicals A21 Radical expressions can be combined (added or subtracted) if they are like radicals—that is, if they have the same index and radicand. For instance, "2, 3"2, and 12"2 are like radicals, but "3 and "2 are unlike radicals. To determine whether two radicals can be combined, you should first simplify each radical. Example 12 Combining Radicals a. 2"48 ! 3"27 # 2"16 * 3 ! 3"9 * 3 Find square factors. Find square roots and multiply by coefficients. # 8"3 ! 9"3 # (8 ! 9'"3 # ! "3 3 3 4 3 8 3 27 b. "16x ! "54x # " * 2x ! " 3 2x ! 3x" 3 2x # 2" 3 # (2 ! 3x' "2x Combine like terms. Simplify. * x3 * 2x Find cube factors. Find cube roots. Combine like terms. Now try Exercise 87. Rationalizing Denominators and Numerators To rationalize a denominator or numerator of the form a ! b"m or a ( b"m, multiply both numerator and denominator by a conjugate: a ( b"m and a ! b"m are conjugates of each other. If a # 0, then the rationalizing factor for "m is itself, "m. For cube roots, choose a rationalizing factor that generates a perfect cube. Example 13 Rationalizing Single-Term Denominators Rationalize the denominator of each expression. 5 2 a. b. 3 " " 2 3 5 Solution a. 5 2"3 # 5 "3 *" 2"3 3 " 5 3 # 2(3' 5"3 # 6 3 52 2 2 " b. 3 # 3 * 3 2 "5 "5 "5 3 52 2" # 3 3 "5 3 25 2" # 5 "3 is rationalizing factor. Multiply. Simplify. 3 52 is rationalizing factor. " Multiply. Simplify. Now try Exercise 95. A22 Appendix A Review of Fundamental Concepts of Algebra Example 14 Rationalizing a Denominator with Two Terms 2 2 # 3 ( "7 3 ( "7 Multiply numerator and denominator by conjugate of denominator. 3 ! "7 * 3 ! "7 # 2(3 ! "7 ' 3(3' ( 3(!"7 ' ( "7(3' ! ("7 '("7 ' Use Distributive Property. # 2(3 ! "7 ' (3'2 ! ("7 '2 Simplify. # 2(3 ! "7 ' 9!7 Square terms of denominator. # 2(3 ! "7 ' # 3 ! "7 2 Simplify. Now try Exercise 97. Sometimes it is necessary to rationalize the numerator of an expression. For instance, in Appendix A.4 you will use the technique shown in the next example to rationalize the numerator of an expression from calculus. WARNING / CAUTION Do not confuse the expression "5 ( "7 with the expression "5 ( 7. In general, "x ( y does not equal "x ( "y. Similarly, "x 2 ( y 2 does not equal x ( y. Example 15 "5 ! "7 2 Rationalizing a Numerator # "5 ! "7 "5 ( "7 * "5 ( "7 2 Multiply numerator and denominator by conjugate of numerator. # ("5 '2 ! ("7 '2 2("5 ( "7' Simplify. # 5!7 2("5 ( "7 ' Square terms of numerator. # !2 !1 # 2("5 ( "7 ' "5 ( "7 Simplify. Now try Exercise 101. Rational Exponents Definition of Rational Exponents If a is a real number and n is a positive integer such that the principal nth root of a exists, then a1)n is defined as n a, where 1)n is the rational exponent of a. a1)n # " Moreover, if m is a positive integer that has no common factor with n, then n a a m)n # (a1)n'm # (" ' m The symbol and n a m. a m)n # (a m'1)n # " indicates an example or exercise that highlights algebraic techniques specifically used in calculus. Appendix A.2 WARNING / CAUTION Rational exponents can be tricky, and you must remember that the expression bm)n is not n b is a real defined unless " number. This restriction produces some unusual-looking results. For instance, the number (!8'1)3 is defined because 3 " !8 # !2, but the number (!8'2)6 is undefined because 6 !8 is not a real number. " Exponents and Radicals A23 The numerator of a rational exponent denotes the power to which the base is raised, and the denominator denotes the index or the root to be taken. Power Index n b n bm b m)n # (" ' #" m When you are working with rational exponents, the properties of integer exponents still apply. For instance, 21)221)3 # 2(1)2)((1)3) # 25)6. Example 16 Changing From Radical to Exponential Form a. "3 # 31)2 2 (3xy'5 # (3xy'5)2 b. "(3xy'5 # " 4 x3 # (2x'(x3)4' # 2x1((3)4) # 2x7)4 c. 2x " Now try Exercise 103. Example 17 T E C H N O LO G Y Changing From Exponential to Radical Form a. (x 2 ( y 2'3)2 # ("x 2 ( y 2 ' # "(x 2 ( y 2'3 3 > There are four methods of evaluating radicals on most graphing calculators. For square roots, you can use the square root key " . For cube roots, you can use the cube root key "3 . For other roots, you can first convert the radical to exponential form and then use the exponential key , or you can use the xth root key x " (or menu choice). Consult the user’s guide for your calculator for specific keystrokes. 4 y3z b. 2y3)4z1)4 # 2( y3z'1)4 # 2 " c. a!3)2 # 1 1 # a3)2 "a3 5 x d. x 0.2 # x1)5 # " Now try Exercise 105. Rational exponents are useful for evaluating roots of numbers on a calculator, for reducing the index of a radical, and for simplifying expressions in calculus. Example 18 Simplifying with Rational Exponents 5 !32 a. (!32'!4)5 # (" ' !4 # (!2'!4 # 1 1 # (!2'4 16 b. (!5x5)3'(3x!3)4' # !15x(5)3)!(3)4) # !15x11)12, c. 9 a3 " d. 3 " 6 125 # " 6 (5'3 # 53)6 # 51)2 # "5 " 125 # " # a3)9 # a1)3 # 3 a " x$0 Reduce index. e. (2x ! 1'4)3(2x ! 1'!1)3 # (2x ! 1'(4)3)!(1)3) # 2x ! 1, x$ 1 2 Now try Exercise 115. The expression in Example 18(e) is not defined when x # -2 * 12 ! 1. !1)3 is not a real number. # (0'!1)3 1 because 2 A24 Appendix A A.2 Review of Fundamental Concepts of Algebra EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises. VOCABULARY: Fill in the blanks. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. In the exponential form an, n is the ________ and a is the ________. A convenient way of writing very large or very small numbers is called ________ ________. One of the two equal factors of a number is called a __________ __________ of the number. n The ________ ________ ________ of a number a is the nth root that has the same sign as a, and is denoted by " a. n In the radical form " a, the positive integer n is called the ________ of the radical and the number a is called the ________. When an expression involving radicals has all possible factors removed, radical-free denominators, and a reduced index, it is in ________ ________. Radical expressions can be combined (added or subtracted) if they are ________ ________. The expressions a ( b"m and a ! b"m are ________ of each other. The process used to create a radical-free denominator is known as ________ the denominator. In the expression bm)n, m denotes the ________ to which the base is raised and n denotes the ________ or root to be taken. SKILLS AND APPLICATIONS In Exercises 11–18, evaluate each expression. In Exercises 31–38, simplify each expression. 11. (a) 32 * 3 55 12. (a) 2 5 13. (a) (33'0 14. (a) (23 * 32'2 (b) 3 * 33 32 (b) 4 3 (b) !32 3 2 (b) (! 35 ' (53 ' 31. (a) (!5z'3 32. (a) (3x'2 3 15. (a) !4 3 (b) 48(!4)!3 4 * 3!2 2!2 * 3!1 17. (a) 2!1 ( 3!1 18. (a) 3!1 ( 2!2 16. (a) 33. (a) 6y 2(2y0'2 34. (a) (!z'3(3z4' (b) (!2'0 (b) (2!1'!2 (b) (3!2'2 37. (a) +(x2y!2'!1*!1 In Exercises 19 –22, use a calculator to evaluate the expression. (If necessary, round your answer to three decimal places.) 19. (!4' ( ' 36 21. 3 7 3 52 7x 2 x3 r4 36. (a) 6 r 35. (a) 20. ( '( 43 22. !4 3 8!4 ' 103 38. (a) (6x7'0, x$0 39. (a) (x ( 5'0, x $ !5 40. (a) (2x5'0, x $ 0 41. (a) (!2x 2'3(4x3'!1 23. 25. 27. 29. 42. (a) (4y!2'(8y4' 24. 26. 28. 30. 7x!2, x # 4 5(!x'3, x # 3 !3x 4, x # !2 12(!x'3, x # ! 13 - .- . a b (b) .b a. !2 3 !2 (b) (5x2z6'3(5x2z6'!3 In Exercises 39–44, rewrite each expression with positive exponents and simplify. In Exercises 23–30, evaluate the expression for the given value of x. !3x 3, x # 2 6x 0, x # 10 2x 3, x # !3 !20x2, x # ! 12 (b) 5x4(x2' (b) (4x 3'0, x $ 0 3x 5 (b) 3 x 25y8 (b) 10y4 12(x ( y'3 (b) 9(x ( y' 4 3 3 4 (b) y y 43. (a) 3n 44. (a) * 32n x 2 * xn x 3 * xn (b) (2x 2'!2 (b) (z ( 2'!3(z ( 2'!1 x !1 (b) 10 x!3y 4 !3 (b) 5 !2 a b 3 (b) b!2 a a!3 a 3 (b) b!3 b - . - . - .- . - .- . Appendix A.2 In Exercises 45–52, write the number in scientific notation. 45. 47. 49. 50. 51. 10,250.4 46. !7,280,000 !0.000125 48. 0.00052 Land area of Earth: 57,300,000 square miles Light year: 9,460,000,000,000 kilometers Relative density of hydrogen: 0.0000899 gram per cubic centimeter 52. One micron (millionth of a meter): 0.00003937 inch In Exercises 53– 60, write the number in decimal notation. 53. 1.25 + 105 54. !1.801 + 105 55. !2.718 + 10!3 56. 3.14 + 10!4 57. Interior temperature of the sun: 1.5 + 107 degrees Celsius 58. Charge of an electron: 1.6022 + 10!19 coulomb 59. Width of a human hair: 9.0 + 10!5 meter 60. Gross domestic product of the United States in 2007: 1.3743021 + 1013 dollars (Source: U.S. Department of Commerce) In Exercises 61 and 62, evaluate each expression without using a calculator. 61. (a) (2.0 + 109'(3.4 + 10!4' (b) (1.2 + 107'(5.0 + 10!3' 6.0 + 108 62. (a) 3.0 + 10!3 2.5 + 10!3 (b) 5.0 + 102 In Exercises 63 and 64, use a calculator to evaluate each expression. (Round your answer to three decimal places.) - . 0.11 800 365 67,000,000 ( 93,000,000 (b) 0.0052 63. (a) 750 1 ( 64. (a) (9.3 + 106'3(6.1 + 10!4' (b) (2.414 + 104'6 (1.68 + 105'5 In Exercises 65–70, evaluate each expression without using a calculator. 65. 66. 67. 68. (a) (a) (a) (a) "9 271)3 32!3)5 100!3)2 1 !1)3 69. (a) ! 64 125 !1)3 70. (a) ! 27 - . . 3 27 (b) " 8 (b) 363)2 !3)4 (b) (16 81 ' !1)2 (b) (94 ' 1 !2)5 (b) "32 1 !4)3 (b) ! 125 - . - . A25 Exponents and Radicals In Exercises 71–76, use a calculator to approximate the number. (Round your answer to three decimal places.) 71. (a) "57 3 452 72. (a) " 73. (a) (!12.4'!1.8 74. (a) 7 ! (4.1'!3.2 2 75. (a) "4.5 + 109 76. (a) (2.65 + 10!4'1)3 5 !273 (b) " 6 (b) " 125 !2.5 (b) (5"3' (b) - . 13 3 !3)2 - . ! ! 3 2 13)3 3 (b) " 6.3 + 104 (b) "9 + 10!4 In Exercises 77 and 78, use the properties of radicals to simplify each expression. 5 2 5 77. (a) (" ' 78. (a) "12 * "3 5 96x5 (b) " 4 (3x2'4 (b) " In Exercises 79–90, simplify each radical expression. 79. (a) "20 3 16 80. (a) " 27 81. (a) "72x3 82. (a) "54xy4 83. 84. 85. 86. 87. 88. (a) (a) (a) (a) (a) (a) (b) 89. (a) (b) 90. (a) (b) 3 16x5 " 4 3x 4 y 2 " 2"50 ( 12"8 4"27 ! "75 5"x ! 3"x 8"49x ! 14"100x !3"48x 2 ( 7" 75x 2 3"x ( 1 ( 10"x ( 1 7"80x ! 2"125x ! "x 3 ! 7 ( 5"x 3 ! 7 11"245x 3 ! 9"45x 3 3 128 (b) " (b) "75 4 182 (b) z3 32a4 (b) b2 (b) "75x2y!4 5 160x 8z 4 (b) " (b) 10"32 ! 6"18 3 16 ( 3" 3 54 (b) " (b) !2"9y ( 10"y " " In Exercises 91–94, complete the statement with <, #, or >. "113 ! "3 91. "5 ( "3 !"5 ( 3 92. 93. 5!"32 ( 22 94. 5!"32 ( 42 "11 In Exercises 95–98, rationalize the denominator of the expression. Then simplify your answer. 95. 97. 1 96. "3 5 "14 ! 2 98. 8 3 " 2 3 "5 ( "6 A26 Appendix A Review of Fundamental Concepts of Algebra In Exercises 99 –102, rationalize the numerator of the expression. Then simplify your answer. 99. "8 2 "5 ( "3 101. 3 100. t # 0.03+125)2 ! (12 ! h'5)2*, 0 % h % 12 "2 3 "7 ! 3 102. 4 In Exercises 103 –110, fill in the missing form of the expression. Radical Form 103. "2.5 3 64 104. " 105.! 106.! 3 !216 107. " 108.! 4 81 3 109. (" ' 110.! Rational Exponent Form ! ! 811)4 ! (1441)2' ! (!243'1)5 ! 165)4 In Exercises 111–114, perform the operations and simplify. ( ' 21)2x4 x!3 * x1)2 113. 3)2 !1 x *x 111. 2x2 3)2 112. (xy'1)3 5!1)2 * 5x5)2 114. (5x'3)2 115. (a) 6 x3 116. (a) " (b) (x ( 1' 4 (3x2'4 (b) " 6 " 4 In Exercises 117 and 118, write each expression as a single radical. Then simplify your answer. 117. (a) 118. (a) ""32 ""243(x ( 1' (b) (b) 4 2x "" 3 10a7b "" 119. PERIOD OF A PENDULUM The period T (in seconds) of a pendulum is T # 2""L)32, where L is the length of the pendulum (in feet). Find the period of a pendulum whose length is 2 feet. 120. EROSION A stream of water moving at the rate of v feet per second can carry particles of size 0.03"v inches. Find the size of the largest particle that can be carried by a stream flowing at the rate of 34 foot per second. The symbol represents the amount of time t (in seconds) that it will take for the funnel to empty. (a) Use the table feature of a graphing utility to find the times required for the funnel to empty for water heights of h # 0, h # 1, h # 2, . . . h # 12 centimeters. (b) What value does t appear to be approaching as the height of the water becomes closer and closer to 12 centimeters? 122. SPEED OF LIGHT The speed of light is approximately 11,180,000 miles per minute. The distance from the sun to Earth is approximately 93,000,000 miles. Find the time for light to travel from the sun to Earth. EXPLORATION TRUE OR FALSE? In Exercises 123 and 124, determine whether the statement is true or false. Justify your answer. 123. x 4)3y 2)3 In Exercises 115 and 116, reduce the index of each radical. 4 32 " 121. MATHEMATICAL MODELING A funnel is filled with water to a height of h centimeters. The formula indicates an example or exercise that highlights algebraic techniques specifically used in calculus. The symbol indicates an exercise or a part of an exercise in which you are instructed to use a graphing utility. x k(1 # xk x 124. (a n' k # a n k 125. Verify that a0 # 1, a $ 0. (Hint: Use the property of exponents am)a n # am!n.) 126. Explain why each of the following pairs is not equal. (a) (3x'!1 $ 3 x (b) y 3 * y2 $ y6 (c) (a 2b 3'4 $ a6b7 (d) (a ( b'2 $ a 2 ( b2 (e) "4x 2 $ 2x (f) "2 ( "3 $ "5 127. THINK ABOUT IT Is 52.7 + 105 written in scientific notation? Why or why not? 128. List all possible digits that occur in the units place of the square of a positive integer. Use that list to determine whether "5233 is an integer. 129. THINK ABOUT IT Square the real number 5)"3 and note that the radical is eliminated from the denominator. Is this equivalent to rationalizing the denominator? Why or why not? 130. CAPSTONE (a) Explain how to simplify the expression (3x3 y!2'!2. (b) Is the expression or why not? "x4 in simplest form? Why 3 Appendix A.3 A27 Polynomials and Factoring A.3 POLYNOMIALS AND FACTORING What you should learn • Write polynomials in standard form. • Add, subtract, and multiply polynomials. • Use special products to multiply polynomials. • Remove common factors from polynomials. • Factor special polynomial forms. • Factor trinomials as the product of two binomials. • Factor polynomials by grouping. Why you should learn it Polynomials can be used to model and solve real-life problems. For instance, in Exercise 224 on page A37, a polynomial is used to model the volume of a shipping box. Polynomials The most common type of algebraic expression is the polynomial. Some examples are 2x ( 5, 3x 4 ! 7x 2 ( 2x ( 4, and 5x 2y 2 ! xy ( 3. The first two are polynomials in x and the third is a polynomial in x and y. The terms of a polynomial in x have the form ax k, where a is the coefficient and k is the degree of the term. For instance, the polynomial 2x 3 ! 5x 2 ( 1 # 2x 3 ( (!5' x 2 ( (0' x ( 1 has coefficients 2, !5, 0, and 1. Definition of a Polynomial in x Let a0, a1, a2, . . . , an be real numbers and let n be a nonnegative integer. A polynomial in x is an expression of the form an x n ( an!1x n!1 ( . . . ( a1x ( a 0 where an $ 0. The polynomial is of degree n, an is the leading coefficient, and a0 is the constant term. Polynomials with one, two, and three terms are called monomials, binomials, and trinomials, respectively. In standard form, a polynomial is written with descending powers of x. Example 1 Writing Polynomials in Standard Form Standard Form Degree Leading Coefficient !5x 7 ( 4x 2 ( 3x ! 2 !9x 2 ( 4 8 (8 # 8x 0' 7 2 0 !5 !9 8 Polynomial a. 4x 2 ! 5x 7 ! 2 ( 3x b. 4 ! 9x 2 c. 8 Now try Exercise 19. A polynomial that has all zero coefficients is called the zero polynomial, denoted by 0. No degree is assigned to this particular polynomial. For polynomials in more than one variable, the degree of a term is the sum of the exponents of the variables in the term. The degree of the polynomial is the highest degree of its terms. For instance, the degree of the polynomial !2x 3y6 ( 4xy ! x7y 4 is 11 because the sum of the exponents in the last term is the greatest. The leading coefficient of the polynomial is the coefficient of the highest-degree term. Expressions are not polynomials if a variable is underneath a radical or if a polynomial expression (with degree greater than 0) is in the denominator of a term. The following expressions are not polynomials. x 3 ! "3x # x 3 ! (3x'1)2 x2 ( 5 # x 2 ( 5x!1 x The exponent “1)2” is not an integer. The exponent “!1” is not a nonnegative integer. A28 Appendix A Review of Fundamental Concepts of Algebra Operations with Polynomials You can add and subtract polynomials in much the same way you add and subtract real numbers. Simply add or subtract the like terms (terms having the same variables to the same powers) by adding their coefficients. For instance, !3xy 2 and 5xy 2 are like terms and their sum is !3xy 2 ( 5xy 2 # (!3 ( 5' xy 2 # 2xy 2. WARNING / CAUTION When an expression inside parentheses is preceded by a negative sign, remember to distribute the negative sign to each term inside the parentheses, as shown. ! (x 2 ! x ( 3' # !x 2 ( x ! 3 Example 2 Sums and Differences of Polynomials a. (5x 3 ! 7x 2 ! 3' ( (x 3 ( 2x 2 ! x ( 8' # (5x 3 ( x 3' ( (!7x2 ( 2x2' ! x ( (!3 ( 8' # 6x 3 ! 5x 2 ! x ( 5 b. (7x4 ! x 2 ! 4x ( 2' ! (3x4 ! 4x 2 ( 3x' # 7x 4 ! x 2 ! 4x ( 2 ! 3x 4 ( 4x 2 ! 3x #( ! '(( ( 4 2 # 4x ( 3x ! 7x ( 2 7x 4 3x 4 !x2 Group like terms. Combine like terms. Distributive Property ' ( (!4x ! 3x' ( 2 4x2 Group like terms. Combine like terms. Now try Exercise 41. To find the product of two polynomials, use the left and right Distributive Properties. For example, if you treat 5x ( 7 as a single quantity, you can multiply 3x ! 2 by 5x ( 7 as follows. (3x ! 2'(5x ( 7' # 3x(5x ( 7' ! 2(5x ( 7' # (3x'(5x' ( (3x'(7' ! (2'(5x' ! (2'(7' # 15x 2 ( 21x ! 10x ! 14 Product of First terms Product of Outer terms Product of Inner terms Product of Last terms # 15x 2 ( 11x ! 14 Note in this FOIL Method (which can only be used to multiply two binomials) that the outer (O) and inner (I) terms are like terms and can be combined. Example 3 Finding a Product by the FOIL Method Use the FOIL Method to find the product of 2x ! 4 and x ( 5. Solution F O I L (2x ! 4'(x ( 5' # 2x 2 ( 10x ! 4x ! 20 # 2x 2 ( 6x ! 20 Now try Exercise 55. Appendix A.3 Polynomials and Factoring A29 Special Products Some binomial products have special forms that occur frequently in algebra. You do not need to memorize these formulas because you can use the Distributive Property to multiply. However, becoming familiar with these formulas will enable you to manipulate the algebra more quickly. Special Products Let u and v be real numbers, variables, or algebraic expressions. Special Product Sum and Difference of Same Terms (u ( v'(u ! v' # u 2 ! v 2 Example (x ( 4'(x ! 4' # x 2 ! 42 # x 2 ! 16 Square of a Binomial (u ( v' 2 # u 2 ( 2uv ( v 2 (x ( 3' 2 # x 2 ( 2(x'(3' ( 32 # x 2 ( 6x ( 9 (3x ! 2'2 # (3x'2 ! 2(3x'(2' ( 22 # 9x 2 ! 12x ( 4 (u ! v' 2 # u 2 ! 2uv ( v 2 Cube of a Binomial (u ( v'3 # u 3 ( 3u 2v ( 3uv 2 ( v 3 (x ( 2'3 # x 3 ( 3x 2(2' ( 3x(22' (23 # x 3 ( 6x 2 ( 12x ( 8 (x !1'3 # x 3 !3x 2(1' (3x(12' !13 # x 3 ! 3x 2 ( 3x ! 1 (u ! v'3 # u 3 ! 3u 2v ( 3uv 2 ! v 3 Example 4 Special Products Find each product. a. (5x ( 9'(5x ! 9' b. (x ( y ! 2'(x ( y ( 2' Solution a. The product of a sum and a difference of the same two terms has no middle term and takes the form (u ( v'(u ! v' # u 2 ! v 2. (5x ( 9'(5x ! 9' # (5x'2 ! 9 2 # 25x 2 ! 81 b. By grouping x ( y in parentheses, you can write the product of the trinomials as a special product. Difference Sum (x ( y ! 2'(x ( y ( 2' # +(x ( y' ! 2*+(x ( y' ( 2* # (x ( y' 2 ! 22 2 2 # x ( 2xy ( y ! 4 Now try Exercise 75. Sum and difference of same terms A30 Appendix A Review of Fundamental Concepts of Algebra Polynomials with Common Factors The process of writing a polynomial as a product is called factoring. It is an important tool for solving equations and for simplifying rational expressions. Unless noted otherwise, when you are asked to factor a polynomial, you can assume that you are looking for factors with integer coefficients. If a polynomial cannot be factored using integer coefficients, then it is prime or irreducible over the integers. For instance, the polynomial x 2 ! 3 is irreducible over the integers. Over the real numbers, this polynomial can be factored as x 2 ! 3 # (x ( "3 '(x ! "3 '. A polynomial is completely factored when each of its factors is prime. For instance x 3 ! x 2 ( 4x ! 4 # (x ! 1'(x 2 ( 4' Completely factored is completely factored, but x 3 ! x 2 ! 4x ( 4 # (x ! 1'(x 2 ! 4' Not completely factored is not completely factored. Its complete factorization is x 3 ! x 2 ! 4x ( 4 # (x ! 1'(x ( 2'(x ! 2'. The simplest type of factoring involves a polynomial that can be written as the product of a monomial and another polynomial. The technique used here is the Distributive Property, a(b ( c' # ab ( ac, in the reverse direction. ab ( ac # a(b ( c' a is a common factor. Removing (factoring out) any common factors is the first step in completely factoring a polynomial. Example 5 Removing Common Factors Factor each expression. a. 6x 3 ! 4x b. !4x 2 ( 12x ! 16 c. (x ! 2'(2x' ( (x ! 2'(3' Solution a. 6x 3 ! 4x # 2x(3x 2' ! 2x(2' 2x is a common factor. # 2x(3x 2 ! 2' b. !4x 2 ( 12x ! 16 # !4(x 2' ( (!4'(!3x' ( (!4'4 !4 is a common factor. # !4(x 2 ! 3x ( 4' c. (x ! 2'(2x' ( (x ! 2'(3' # (x ! 2'(2x ( 3' Now try Exercise 99. (x ! 2' is a common factor. Appendix A.3 Polynomials and Factoring A31 Factoring Special Polynomial Forms Some polynomials have special forms that arise from the special product forms on page A29. You should learn to recognize these forms so that you can factor such polynomials easily. Factoring Special Polynomial Forms Factored Form Difference of Two Squares Example u 2 ! v 2 # (u ( v'(u ! v' 9x 2 ! 4 # (3x' 2 ! 2 2 # (3x ( 2'(3x ! 2' Perfect Square Trinomial u 2 ( 2uv ( v 2 # (u ( v' 2 x 2 ( 6x ( 9 # x 2 ( 2(x'(3' ( 32 # (x ( 3'2 u 2 ! 2uv ( v 2 # (u ! v' 2 x 2 ! 6x ( 9 # x 2 ! 2(x'(3' ( 32 # (x ! 3'2 Sum or Difference of Two Cubes u 3 ( v 3 # (u ( v'(u 2 ! uv ( v 2' x 3 ( 8 # x 3 ( 23 # (x ( 2'(x 2 ! 2x ( 4' u3 ! v3 # (u ! v'(u2 ( uv ( v 2' 27x3 ! 1 # (3x' 3 ! 13 # (3x ! 1'(9x 2 ( 3x ( 1' One of the easiest special polynomial forms to factor is the difference of two squares. The factored form is always a set of conjugate pairs. u 2 ! v 2 # (u ( v'(u ! v' Difference Conjugate pairs Opposite signs To recognize perfect square terms, look for coefficients that are squares of integers and variables raised to even powers. Example 6 In Example 6, note that the first step in factoring a polynomial is to check for any common factors. Once the common factors are removed, it is often possible to recognize patterns that were not immediately obvious. Removing a Common Factor First 3 ! 12x 2 # 3(1 ! 4x 2' 3 is a common factor. # 3+1 ! (2x' * 2 2 # 3(1 ( 2x'(1 ! 2x' Difference of two squares Now try Exercise 113. Example 7 Factoring the Difference of Two Squares a. (x ( 2'2 ! y 2 # +(x ( 2' ( y*+(x ( 2' ! y* # (x ( 2 ( y'(x ( 2 ! y' b. 16x 4 ! 81 # (4x 2'2 ! 92 # (4x 2 ( 9'(4x 2 ! 9' # (4x2 ( 9'+(2x'2 ! 32* # (4x2 ( 9'(2x ( 3'(2x ! 3' Now try Exercise 117. Difference of two squares Difference of two squares A32 Appendix A Review of Fundamental Concepts of Algebra A perfect square trinomial is the square of a binomial, and it has the following form. u 2 ( 2uv ( v 2 # (u ( v' 2 or u 2 ! 2uv ( v 2 # (u ! v' 2 Like signs Like signs Note that the first and last terms are squares and the middle term is twice the product of u and v. Example 8 Factoring Perfect Square Trinomials Factor each trinomial. a. x 2 ! 10x ( 25 b. 16x 2 ( 24x ( 9 Solution a. x 2 ! 10x ( 25 # x 2 ! 2(x'(5' ( 5 2 # (x ! 5'2 b. 16x2 ( 24x ( 9 # (4x'2 ( 2(4x'(3' ( 32 # (4x ( 3'2 Now try Exercise 123. The next two formulas show the sums and differences of cubes. Pay special attention to the signs of the terms. Like signs Like signs u 3 ( v 3 # (u ( v'(u 2 ! uv ( v 2' u 3 ! v 3 # (u ! v'(u 2 ( uv ( v 2' Unlike signs Example 9 Unlike signs Factoring the Difference of Cubes Factor x 3 ! 27. Solution x3 ! 27 # x3 ! 33 Rewrite 27 as 33. # (x ! 3'(x 2 ( 3x ( 9' Factor. Now try Exercise 133. Example 10 Factoring the Sum of Cubes a. y 3 ( 8 # y 3 ( 23 Rewrite 8 as 23. # ( y ( 2'( y 2 ! 2y ( 4' b. 3( x3 ( 64' # 3( x3 ( ' 43 # 3(x ( 4'(x 2 ! 4x ( 16' Now try Exercise 135. Factor. Rewrite 64 as 43. Factor. Appendix A.3 Polynomials and Factoring A33 Trinomials with Binomial Factors To factor a trinomial of the form ax 2 ( bx ( c, use the following pattern. Factors of a ax2 ( bx ( c # (!x ( !'(!x ( !' Factors of c The goal is to find a combination of factors of a and c such that the outer and inner products add up to the middle term bx. For instance, in the trinomial 6x 2 ( 17x ( 5, you can write all possible factorizations and determine which one has outer and inner products that add up to 17x. (6x ( 5'(x ( 1', (6x ( 1'(x ( 5', (2x ( 1'(3x ( 5', (2x ( 5'(3x ( 1' You can see that (2x ( 5'(3x ( 1' is the correct factorization because the outer (O) and inner (I) products add up to 17x. F O I L O(I (2x ( 5'(3x ( 1' # 6x 2 ( 2x ( 15x ( 5 # 6x2 ( 17x ( 5 Example 11 Factoring a Trinomial: Leading Coefficient Is 1 Factor x 2 ! 7x ( 12. Solution The possible factorizations are (x ! 2'(x ! 6', (x ! 1'(x ! 12', and (x ! 3'(x ! 4'. Testing the middle term, you will find the correct factorization to be x 2 ! 7x ( 12 # (x ! 3'(x ! 4'. Now try Exercise 145. Example 12 Factoring a Trinomial: Leading Coefficient Is Not 1 Factor 2x 2 ( x ! 15. Solution Factoring a trinomial can involve trial and error. However, once you have produced the factored form, it is an easy matter to check your answer. For instance, you can verify the factorization in Example 11 by multiplying out the expression (x ! 3'(x ! 4' to see that you obtain the original trinomial, x2 ! 7x ( 12. The eight possible factorizations are as follows. (2x ! 1'(x ( 15' (2x ( 1'(x ! 15' (2x ! 3'(x ( 5' (2x ( 3'(x ! 5' (2x ! 5'(x ( 3' (2x ( 5'(x ! 3' (2x ! 15'(x ( 1' (2x ( 15'(x ! 1' Testing the middle term, you will find the correct factorization to be 2x 2 ( x ! 15 # (2x ! 5'(x ( 3'. Now try Exercise 153. O ( I # 6x ! 5x # x A34 Appendix A Review of Fundamental Concepts of Algebra Factoring by Grouping Sometimes polynomials with more than three terms can be factored by a method called factoring by grouping. It is not always obvious which terms to group, and sometimes several different groupings will work. Example 13 Factoring by Grouping Use factoring by grouping to factor x 3 ! 2x 2 ! 3x ( 6. Another way to factor the polynomial in Example 13 is to group the terms as follows. x3 ! 2x2 Solution x 3 ! 2x 2 ! 3x ( 6 # (x 3 ! 2x 2' ! (3x ! 6' ! 3x ( 6 # (x3 ! 3x' ! (2x2 ! 6' # x(x ! 3' ! 2(x ! 3' 2 Group terms. # x 2(x ! 2' ! 3(x ! 2' Factor each group. # (x ! 2'(x 2 ! 3' Distributive Property Now try Exercise 161. 2 # (x2 ! 3'(x ! 2' As you can see, you obtain the same result as in Example 13. Factoring a trinomial can involve quite a bit of trial and error. Some of this trial and error can be lessened by using factoring by grouping. The key to this method of factoring is knowing how to rewrite the middle term. In general, to factor a trinomial ax2 ( bx ( c by grouping, choose factors of the product ac that add up to b and use these factors to rewrite the middle term. This technique is illustrated in Example 14. Example 14 Factoring a Trinomial by Grouping Use factoring by grouping to factor 2x 2 ( 5x ! 3. Solution In the trinomial 2x 2 ( 5x ! 3, a # 2 and c # !3, which implies that the product ac is !6. Now, !6 factors as (6'(!1' and 6 ! 1 # 5 # b. So, you can rewrite the middle term as 5x # 6x ! x. This produces the following. 2x 2 ( 5x ! 3 # 2x 2 ( 6x ! x ! 3 #( 2x 2 ( 6x' ! (x ( 3' Rewrite middle term. Group terms. # 2x(x ( 3' ! (x ( 3' Factor groups. # (x ( 3'(2x ! 1' Distributive Property So, the trinomial factors as 2x 2 ( 5x ! 3 # (x ( 3'(2x ! 1'. Now try Exercise 167. Guidelines for Factoring Polynomials 1. Factor out any common factors using the Distributive Property. 2. Factor according to one of the special polynomial forms. 3. Factor as ax2 ( bx ( c # (mx ( r'(nx ( s'. 4. Factor by grouping. Appendix A.3 A.3 EXERCISES Polynomials and Factoring A35 See www.CalcChat.com for worked-out solutions to odd-numbered exercises. VOCABULARY: Fill in the blanks. 1. For the polynomial an x n ( an!1 x n!1 ( . . . ( a1x ( a0, an $ 0, the degree is ________, the leading coefficient is ________, and the constant term is ________. 2. A polynomial in x in standard form is written with ________ powers of x. 3. A polynomial with one term is called a ________, while a polynomial with two terms is called a ________, and a polynomial with three terms is called a ________. 4. To add or subtract polynomials, add or subtract the ________ ________ by adding their coefficients. 5. The letters in “FOIL” stand for the following. F ________ O ________ I ________ L ________ 6. The process of writing a polynomial as a product is called ________. 7. A polynomial is ________ ________ when each of its factors is prime. 8. A polynomial u2 ( 2uv ( v2 is called a ________ ________ ________. SKILLS AND APPLICATIONS In Exercises 9–14, match the polynomial with its description. [The polynomials are labeled (a), (b), (c), (d), (e), and (f).] (a) (c) (e) 9. 10. 11. 12. 13. 14. 3x2 x3 ( 3x2 ( 3x ( 1 !3x5 ( 2x3 ( x (b) 1 ! 2x3 (d) 12 2 (f) 3 x4 ( x2 ( 10 A polynomial of degree 0 A trinomial of degree 5 A binomial with leading coefficient !2 A monomial of positive degree 2 A trinomial with leading coefficient 3 A third-degree polynomial with leading coefficient 1 In Exercises 15–18, write a polynomial that fits the description. (There are many correct answers.) 15. A third-degree polynomial with leading coefficient !2 16. A fifth-degree polynomial with leading coefficient 6 17. A fourth-degree binomial with a negative leading coefficient 18. A third-degree binomial with an even leading coefficient In Exercises 19–30, (a) write the polynomial in standard form, (b) identify the degree and leading coefficient of the polynomial, and (c) state whether the polynomial is a monomial, a binomial, or a trinomial. 19. 21. 23. 25. 27. 29. 14x ! 12 x 5 x2 ! 4 ! 3x4 3 ! x6 3 1 ( 6x 4 ! 4x 5 4x 3y 20. 22. 24. 26. 28. 30. 2x 2 ! x ( 1 7x !y ( 25y2 ( 1 !8 ( t2 3 ( 2x !x 5y ( 2x 2y 2 ( xy 4 In Exercises 31–36, determine whether the expression is a polynomial. If so, write the polynomial in standard form. 31. 2x ! 3x 3 ( 8 3x ( 4 33. x 2 35. y ! y 4 ( y 3 32. 5x4 ! 2x2 ( x!2 34. x 2 ( 2x ! 3 2 36. y4 ! "y In Exercises 37–54, perform the operation and write the result in standard form. 37. 38. 39. 40. 41. 42. 43. 44. 45. 47. 49. 51. 53. (6x ( 5' ! (8x ( 15' (2x 2 ( 1' ! (x 2 ! 2x ( 1' ! (t3 ! 1' ( (6t3 ! 5t' ! (5x 2 ! 1' ! (!3x 2 ( 5' (15x 2 ! 6' ! (!8.3x 3 ! 14.7x 2 ! 17' (15.6w4 ! 14w ! 17.4' ! (16.9w4 ! 9.2w ( 13' 5z ! +3z ! (10z ( 8'* ( y 3 ( 1' ! +( y 2 ( 1' ( (3y ! 7'* 3x(x 2 ! 2x ( 1' 46. y 2(4y 2 ( 2y ! 3' !5z(3z ! 1' 48. (!3x'(5x ( 2' 3 (1 ! x '(4x' 50. !4x(3 ! x 3' (1.5t2 ( 5'(!3t' 52. (2 ! 3.5y'(2y 3' 3 !2x(0.1x ( 17' 54. 6y(5 ! 8 y' In Exercises 55–92, multiply or find the special product. 55. 57. 59. 61. 63. (x ( 3'(x ( 4' (3x ! 5'(2x ( 1' (x ( 10'(x ! 10' (x ( 2y'(x ! 2y' (2x ( 3' 2 56. 58. 60. 62. 64. (x ! 5'(x ( 10' (7x ! 2'(4x ! 3' (2x ( 3'(2x ! 3' (4a ( 5b'(4a ! 5b' (5 ! 8x' 2 A36 65. 67. 69. 71. 72. 73. 74. 75. 76. 77. 79. 81. 83. 85. 87. 89. 90. 91. 92. Appendix A Review of Fundamental Concepts of Algebra (x ( 1' 3 66. (x ! 2' 3 (2x ! y' 3 68. (3x ( 2y' 3 3 2 (4x ! 3' 70. (8x ( 3'2 (x 2 ! x ( 1'(x 2 ( x ( 1' (x 2 ( 3x ! 2'(x 2 ! 3x ! 2' (!x2 ( x ! 5'(3x2 ( 4x ( 1' (2x2 ! x ( 4'(x2 ( 3x ( 2' +(m ! 3' ( n*+(m ! 3' ! n* +(x ! 3y' ( z*+(x ! 3y' ! z* +(x ! 3' ( y*2 78. +(x ( 1' ! y*2 2 2 (2r ! 5'(2r ( 5' 80. (3a 3 ! 4b2'(3a 3 ( 4b2' 2 2 82. (35 t ( 4' (14 x ! 5' 84. (3x ( 16 '(3x ! 16 ' (15 x ! 3'(15 x ( 3' (2.4x ( 3'2 86. (1.8y ! 5'2 (1.5x ! 4'(1.5x ( 4' 88. (2.5y ( 3'(2.5y ! 3' 5x(x ( 1' ! 3x(x ( 1' (2x ! 1'(x ( 3' ( 3(x ( 3' (u ( 2'(u ! 2'(u 2 ( 4' (x ( y'(x ! y'(x 2 ( y 2' In Exercises 93–96, find the product. (The expressions are not polynomials, but the formulas can still be used.) 93. ("x ( "y'("x ! "y' 94. (5 ( "x'(5 ! "x' 2 2 95. (x ! "5 ' 96. (x ( "3 ' In Exercises 97–104, factor out the common factor. 97. 99. 101. 103. 4x ( 16 98. 5y ! 30 2x 3 ! 6x 100. 3z3 ! 6z2 ( 9z 3x(x ! 5' ( 8(x ! 5' 102. 3x(x ( 2' ! 4(x ( 2' (x ( 3'2 ! 4(x ( 3' 104. (5x ! 4'2 ( (5x ! 4' In Exercises 105–110, find the greatest common factor such that the remaining factors have only integer coefficients. 105. 12 x ( 4 106. 13 y ( 5 107. 12 x 3 ( 2x 2 ! 5x 108. 13 y 4 ! 5y 2 ( 2y 109. 23 x(x ! 3' ! 4(x ! 3' 110. 45 y( y ( 1' ! 2( y ( 1' In Exercises 111–120, completely factor the difference of two squares. 111. 113. 115. 117. 119. x2 ! 81 48y2 ! 27 16x 2 ! 19 (x ! 1' 2 ! 4 9u2 ! 4v 2 112. 114. 116. 118. 120. x 2 ! 64 50 ! 98z2 4 2 25 y ! 64 25 ! (z ( 5' 2 25x 2 ! 16y 2 In Exercises 121–132, factor the perfect square trinomial. 121. 123. 125. 127. 129. 131. x 2 ! 4x ( 4 4t 2 ( 4t ( 1 25y 2 ! 10y ( 1 9u2 ( 24uv ( 16v 2 x 2 ! 43x ( 49 4x2 ! 43 x ( 19 122. 124. 126. 128. 130. 132. x 2 ( 10x ( 25 9x 2 ! 12x ( 4 36y 2 ! 108y ( 81 4x 2 ! 4xy ( y 2 z 2 ( z ( 14 1 9y2 ! 32 y ( 16 In Exercises 133–144, factor the sum or difference of cubes. 133. 135. 137. 139. 141. 143. x3 ! 8 y 3 ( 64 8 x3 ! 27 8t 3 ! 1 u3 ( 27v 3 (x ( 2'3 ! y3 134. 136. 138. 140. 142. 144. x 3 ! 27 z 3 ( 216 8 y3 ( 125 27x 3 ( 8 64x 3 ! y 3 (x ! 3y'3 ! 8z3 In Exercises 145–158, factor the trinomial. 145. 147. 149. 151. 153. 155. 157. x2 ( x ! 2 s 2 ! 5s ( 6 20 ! y ! y 2 x 2 ! 30x ( 200 3x 2 ! 5x ( 2 5x 2 ( 26x ( 5 !9z 2 ( 3z ( 2 146. 148. 150. 152. 154. 156. 158. x 2 ( 5x ( 6 t2 ! t ! 6 24 ( 5z ! z 2 x 2 ! 13x ( 42 2x 2 ! x ! 1 12x 2 ( 7x ( 1 !5u 2 ! 13u ( 6 In Exercises 159–166, factor by grouping. 159. 161. 163. 165. x 3 ! x 2 ( 2x ! 2 2x 3 ! x 2 ! 6x ( 3 6 ( 2x ! 3x3 ! x4 6x 3 ! 2x ( 3x 2 ! 1 160. 162. 164. 166. x 3 ( 5x 2 ! 5x ! 25 5x 3 ! 10x 2 ( 3x ! 6 x 5 ( 2x 3 ( x 2 ( 2 8x 5 ! 6x 2 ( 12x 3 ! 9 In Exercises 167–172, factor the trinomial by grouping. 167. 3x 2 ( 10x ( 8 169. 6x 2 ( x ! 2 171. 15x 2 ! 11x ( 2 168. 2x 2 ( 9x ( 9 170. 6x 2 ! x ! 15 172. 12x2 ! 13x ( 1 In Exercises 173–206, completely factor the expression. 173. 175. 177. 179. 181. 183. 185. 6x 2 ! 54 x3 ! x2 x3 ! 16x x 2 ! 2x ( 1 1 ! 4x ( 4x 2 2x 2 ( 4x ! 2x 3 2 1 2 81 x ( 9 x ! 8 174. 176. 178. 180. 182. 184. 186. 12x 2 ! 48 x 3 ! 4x 2 x 3 ! 9x 16 ( 6x ! x 2 !9x 2 ( 6x ! 1 13x ( 6 ( 5x 2 1 2 1 1 8 x ! 96 x ! 16 Appendix A.3 3x 3 ( x 2 ( 15x ( 5 188. 5 ! x ( 5x 2 ! x 3 x 4 ! 4x 3 ( x 2 ! 4x 190. 3x3 ( x2 ! 27x ! 9 1 3 3 2 192. 15 x3 ( x2 ! x ! 5 4 x ( 3x ( 4 x ( 9 (t ! 1' 2 ! 49 194. (x 2 ( 1' 2 ! 4x 2 (x 2 ( 8' 2 ! 36x 2 196. 2t 3 ! 16 5x 3 ( 40 198. 4x(2x ! 1' ( (2x ! 1' 2 2 5(3 ! 4x' ! 8(3 ! 4x'(5x ! 1' 2(x ( 1'(x ! 3' 2 ! 3(x ( 1' 2(x ! 3' 7(3x ( 2' 2(1 ! x' 2 ( (3x ( 2'(1 ! x'3 7x(2'(x 2 ( 1'(2x' ! (x2 ( 1' 2(7' 3(x ! 2'2(x ( 1'4 ( (x ! 2' 3(4'(x ( 1' 3 2x(x ! 5' 4 ! x 2(4'(x ! 5' 3 5(x6 ( 1'4(6x5'(3x ( 2'3 ( 3(3x ( 2'2(3'(x6 ( 1'5 x2 206. (x2 ( 1'4 ! (x 2 ( 1'5 2 (a) Write this polynomial in standard form. (b) Use a calculator to evaluate the polynomial for the values of r shown in the table. r 2% 211. (5x ! 1'(3' ! (3x ( 1'(5' (5x ! 1'2 212. (2x ( 3'(4' ! (4x ! 1'(2' (2x ( 3'2 In Exercises 213–216, find all values of b for which the trinomial can be factored. 213. x 2 ( bx ! 15 215. x 2 ( bx ( 50 214. x 2 ( bx ! 12 216. x 2 ( bx ( 24 218. 3x 2 ! 10x ( c 220. 2x 2 ( 9x ( c 221. COST, REVENUE, AND PROFIT An electronics manufacturer can produce and sell x MP3 players per week. The total cost C (in dollars) of producing x MP3 players is C # 73x ( 25,000, and the total revenue R (in dollars) is R # 95x. (a) Find the profit P in terms of x. (b) Find the profit obtained by selling 5000 MP3 players per week. 4% 412% (c) What conclusion can you make from the table? 223. VOLUME OF A BOX A take-out fast-food restaurant is constructing an open box by cutting squares from the corners of a piece of cardboard that is 18 centimeters by 26 centimeters (see figure). The edge of each cut-out square is x centimeters. x x 26 − 2x 18 cm x x x 18 − 2x 26 − 2x 26 cm (a) Find the volume of the box in terms of x. (b) Find the volume when x # 1, x # 2, and x # 3. 224. VOLUME OF A BOX An overnight shipping company is designing a closed box by cutting along the solid lines and folding along the broken lines on the rectangular piece of corrugated cardboard shown in the figure. The length and width of the rectangle are 45 centimeters and 15 centimeters, respectively. 45 cm x In Exercises 217–220, find two integer values of c such that the trinomial can be factored. (There are many correct answers.) 217. 2x 2 ( 5x ( c 219. 3x 2 ! x ( c 312% 15 cm x4(4'(2x ( 1'3(2x' ( (2x ( 1'4(4x3' x3(3'(x2 ( 1'2(2x' ( (x2 ( 1'3(3x2' (2x ! 5'4(3'(5x ! 4'2(5' ( (5x ! 4'3(4'(2x ! 5'3(2' (x2 ! 5'3(2'(4x ( 3'(4' ( (4x ( 3'2(3'(x2 ! 5'2(x2' 3% 1200(1 ( r'3 In Exercises 207–212, completely factor the expression. 207. 208. 209. 210. A37 222. COMPOUND INTEREST After 3 years, an investment of $1200 compounded annually at an interest rate r will yield an amount of 1200(1 ( r'3. 18 − 2x 187. 189. 191. 193. 195. 197. 199. 200. 201. 202. 203. 204. 205. Polynomials and Factoring (a) Find the volume of the shipping box in terms of x. (b) Find the volume when x # 3, x # 5, and x # 7. 225. GEOMETRY Find a polynomial that represents the total number of square feet for the floor plan shown in the figure. x x 14 ft 22 ft A38 Appendix A Review of Fundamental Concepts of Algebra 226. GEOMETRY Find the area of the shaded region in each figure. Write your result as a polynomial in standard form. (a) (b) 2x + 6 x+4 x 2x 8x 12x 6x 9x EXPLORATION GEOMETRIC MODELING In Exercises 227–230, draw a “geometric factoring model” to represent the factorization. For instance, a factoring model for 2x2 # 3x # 1 " (2x # 1'(x # 1' is shown in the figure. x x x x x 1 1 1 x 1 x 227. 228. 229. 230. x 1 x 1 x 1 3x 2 ( 7x ( 2 # (3x ( 1'(x ( 2' x 2 ( 4x ( 3 # (x ( 3'(x ( 1' 2x 2 ( 7x ( 3 # (2x ( 1'(x ( 3' x 2 ( 3x ( 2 # (x ( 2'(x ( 1' GEOMETRY In Exercises 231 and 232, write an expression in factored form for the area of the shaded portion of the figure. 231. 232. x+3 r 4 r+2 5 5 (x 4 + 3) 233. GEOMETRY The cylindrical shell shown in the figure has a volume of V # "R2h ! "r2h. R h r (a) Factor the expression for the volume. (b) From the result of part (a), show that the volume is 2" (average radius'(thickness of the shell'h. 234. CHEMISTRY The rate of change of an autocatalytic chemical reaction is kQx ! kx 2, where Q is the amount of the original substance, x is the amount of substance formed, and k is a constant of proportionality. Factor the expression. TRUE OR FALSE? In Exercises 235–238, determine whether the statement is true or false. Justify your answer. 235. The product of two binomials is always a seconddegree polynomial. 236. The sum of two binomials is always a binomial. 237. The difference of two perfect squares can be factored as the product of conjugate pairs. 238. The sum of two perfect squares can be factored as the binomial sum squared. 239. Find the degree of the product of two polynomials of degrees m and n. 240. Find the degree of the sum of two polynomials of degrees m and n if m < n. 241. THINK ABOUT IT When the polynomial !x3 ( 3x2 ( 2x ! 1 is subtracted from an unknown polynomial, the difference is 5x 2 ( 8. If it is possible, find the unknown polynomial. 242. LOGICAL REASONING Verify that (x ( y'2 is not equal to x 2 ( y 2 by letting x # 3 and y # 4 and evaluating both expressions. Are there any values of x and y for which (x ( y'2 # x 2 ( y 2 ? Explain. 243. Factor x 2n ! y 2n as completely as possible. 244. Factor x 3n ( y 3n as completely as possible. 245. Give an example of a polynomial that is prime with respect to the integers. 246. CAPSTONE A third-degree polynomial and a fourth-degree polynomial are added. (a) Can the sum be a fourth-degree polynomial? Explain or give an example. (b) Can the sum be a second-degree polynomial? Explain or give an example. (c) Can the sum be a seventh-degree polynomial? Explain or give an example. (d) After adding the two polynomials and factoring the sum, you obtain a polynomial that is in factored form. Explain what is meant by saying that a polynomial is in factored form. Appendix A.4 Rational Expressions A39 A.4 RATIONAL EXPRESSIONS What you should learn • Find domains of algebraic expressions. • Simplify rational expressions. • Add, subtract, multiply, and divide rational expressions. • Simplify complex fractions and rewrite difference quotients. Domain of an Algebraic Expression The set of real numbers for which an algebraic expression is defined is the domain of the expression. Two algebraic expressions are equivalent if they have the same domain and yield the same values for all numbers in their domain. For instance, (x ( 1' ( (x ( 2' and 2x ( 3 are equivalent because (x ( 1' ( (x ( 2' # x ( 1 ( x ( 2 #x(x(1(2 Why you should learn it Rational expressions can be used to solve real-life problems. For instance, in Exercise 102 on page A48, a rational expression is used to model the projected numbers of U.S. households banking and paying bills online from 2002 through 2007. # 2x ( 3. Example 1 Finding the Domain of an Algebraic Expression a. The domain of the polynomial 2x 3 ( 3x ( 4 is the set of all real numbers. In fact, the domain of any polynomial is the set of all real numbers, unless the domain is specifically restricted. b. The domain of the radical expression "x ! 2 is the set of real numbers greater than or equal to 2, because the square root of a negative number is not a real number. c. The domain of the expression x(2 x!3 is the set of all real numbers except x # 3, which would result in division by zero, which is undefined. Now try Exercise 7. The quotient of two algebraic expressions is a fractional expression. Moreover, the quotient of two polynomials such as 1 , x 2x ! 1 , x(1 or x2 ! 1 x2 ( 1 is a rational expression. Simplifying Rational Expressions Recall that a fraction is in simplest form if its numerator and denominator have no factors in common aside from ± 1. To write a fraction in simplest form, divide out common factors. a b * c # a, *c b c$0 A40 Appendix A Review of Fundamental Concepts of Algebra The key to success in simplifying rational expressions lies in your ability to factor polynomials. When simplifying rational expressions, be sure to factor each polynomial completely before concluding that the numerator and denominator have no factors in common. Example 2 WARNING / CAUTION In Example 2, do not make the mistake of trying to simplify further by dividing out terms. x(6 x(6 #x(2 # 3 3 Remember that to simplify fractions, divide out common factors, not terms. Write Simplifying a Rational Expression x 2 ( 4x ! 12 in simplest form. 3x ! 6 Solution x2 ( 4x ! 12 (x ( 6'(x ! 2' # 3x ! 6 3(x ! 2' # x(6 , 3 x$2 Factor completely. Divide out common factors. Note that the original expression is undefined when x # 2 (because division by zero is undefined). To make sure that the simplified expression is equivalent to the original expression, you must restrict the domain of the simplified expression by excluding the value x # 2. Now try Exercise 33. Sometimes it may be necessary to change the sign of a factor by factoring out (!1' to simplify a rational expression, as shown in Example 3. Example 3 Write Simplifying Rational Expressions 12 ( x ! x2 in simplest form. 2x2 ! 9x ( 4 Solution 12 ( x ! x2 (4 ! x'(3 ( x' # 2x2 ! 9x ( 4 (2x ! 1'(x ! 4' # ! (x ! 4'(3 ( x' (2x ! 1'(x ! 4' #! 3(x , x$4 2x ! 1 Factor completely. (4 ! x' # ! (x ! 4' Divide out common factors. Now try Exercise 39. In this text, when a rational expression is written, the domain is usually not listed with the expression. It is implied that the real numbers that make the denominator zero are excluded from the expression. Also, when performing operations with rational expressions, this text follows the convention of listing by the simplified expression all values of x that must be specifically excluded from the domain in order to make the domains of the simplified and original expressions agree. In Example 3, for instance, the restriction x $ 4 is listed with the simplified expression 1 to make the two domains agree. Note that the value x # 2 is excluded from both domains, so it is not necessary to list this value. Appendix A.4 Rational Expressions A41 Operations with Rational Expressions To multiply or divide rational expressions, use the properties of fractions discussed in Appendix A.1. Recall that to divide fractions, you invert the divisor and multiply. Example 4 Multiplying Rational Expressions 2x2 ( x ! 6 x2 ( 4x ! 5 * x3 ! 3x2 ( 2x (2x ! 3'(x ( 2' # 4x2 ! 6x (x ( 5'(x ! 1' # * x(x ! 2'(x ! 1' 2x(2x ! 3' (x ( 2'(x ! 2' , x $ 0, x $ 1, x $ 32 2(x ( 5' Now try Exercise 53. In Example 4, the restrictions x $ 0, x $ 1, and x $ 32 are listed with the simplified expression in order to make the two domains agree. Note that the value x # !5 is excluded from both domains, so it is not necessary to list this value. Example 5 Dividing Rational Expressions x 3 ! 8 x 2 ( 2x ( 4 x 3 ! 8 ) # 2 x2 ! 4 x3 ( 8 x !4 # x3 ( 8 * x 2 ( 2x ( 4 Invert and multiply. (x ! 2'(x2 ( 2x ( 4' (x ( 2'(x2 ! 2x ( 4' * (x2 ( 2x ( 4' (x ( 2'(x ! 2' # x 2 ! 2x ( 4, x $ ± 2 Divide out common factors. Now try Exercise 55. To add or subtract rational expressions, you can use the LCD (least common denominator) method or the basic definition a c ad ± bc ± # , b d bd b $ 0, d $ 0. Basic definition This definition provides an efficient way of adding or subtracting two fractions that have no common factors in their denominators. Example 6 WARNING / CAUTION When subtracting rational expressions, remember to distribute the negative sign to all the terms in the quantity that is being subtracted. Subtracting Rational Expressions x 2 x(3x ( 4' ! 2(x ! 3' ! # x ! 3 3x ( 4 (x ! 3'(3x ( 4' Basic definition # 3x 2 ( 4x ! 2x ( 6 (x ! 3'(3x ( 4' Distributive Property # 3x 2 ( 2x ( 6 (x ! 3'(3x ( 4' Combine like terms. Now try Exercise 65. A42 Appendix A Review of Fundamental Concepts of Algebra For three or more fractions, or for fractions with a repeated factor in the denominators, the LCD method works well. Recall that the least common denominator of several fractions consists of the product of all prime factors in the denominators, with each factor given the highest power of its occurrence in any denominator. Here is a numerical example. 1 3 2 1*2 3*3 2*4 ( ! # ( ! 6 4 3 6*2 4*3 3*4 # 2 9 8 ( ! 12 12 12 # 3 12 # 1 4 The LCD is 12. Sometimes the numerator of the answer has a factor in common with the denominator. In such cases the answer should be simplified. For instance, in the 3 example above, 12 was simplified to 14. Example 7 Combining Rational Expressions: The LCD Method Perform the operations and simplify. 3 2 x(3 ! ( 2 x!1 x x !1 Solution Using the factored denominators (x ! 1', x, and (x ( 1'(x ! 1', you can see that the LCD is x(x ( 1'(x ! 1'. 3 2 x(3 ! ( x!1 x (x ( 1'(x ! 1' # 3(x'(x ( 1' 2(x ( 1'(x ! 1' (x ( 3'(x' ! ( x(x ( 1'(x ! 1' x(x ( 1'(x ! 1' x(x ( 1'(x ! 1' # 3(x'(x ( 1' ! 2(x ( 1'(x ! 1' ( (x ( 3'(x' x(x ( 1'(x ! 1' # 3x 2 ( 3x ! 2x 2 ( 2 ( x 2 ( 3x x(x ( 1'(x ! 1' Distributive Property # 3x 2 ! 2x 2 ( x 2 ( 3x ( 3x ( 2 x(x ( 1'(x ! 1' Group like terms. # 2x2 ( 6x ( 2 x(x ( 1'(x ! 1' Combine like terms. # 2(x 2 ( 3x ( 1' x(x ( 1'(x ! 1' Factor. Now try Exercise 67. Appendix A.4 Rational Expressions A43 Complex Fractions and the Difference Quotient Fractional expressions with separate fractions in the numerator, denominator, or both are called complex fractions. Here are two examples. -x. -x. 1 x2 ( 1 1 and -x 2 1 (1 . To simplify a complex fraction, combine the fractions in the numerator into a single fraction and then combine the fractions in the denominator into a single fraction. Then invert the denominator and multiply. Example 8 Simplifying a Complex Fraction 2 ! 3(x' x # 1 1(x ! 1' ! 1 1! x!1 x!1 2 - x ! 3. - 1 2 . 1 2 - x . # x!2 -x ! 1. Combine fractions. 2 ! 3x Simplify. # 2 ! 3x x x!1 # (2 ! 3x'(x ! 1' , x$1 x(x ! 2' *x!2 Invert and multiply. Now try Exercise 73. Another way to simplify a complex fraction is to multiply its numerator and denominator by the LCD of all fractions in its numerator and denominator. This method is applied to the fraction in Example 8 as follows. - x ! 3. - x ! 3. 2 - 1 1! x!1 2 # . - 1 1! x!1 x(x ! 1' . * x(x ! 1' -2 !x 3x. * x(x ! 1' # -xx !! 21. * x(x ! 1' # (2 ! 3x'(x ! 1' , x$1 x(x ! 2' LCD is x(x ! 1'. A44 Appendix A Review of Fundamental Concepts of Algebra The next three examples illustrate some methods for simplifying rational expressions involving negative exponents and radicals. These types of expressions occur frequently in calculus. To simplify an expression with negative exponents, one method is to begin by factoring out the common factor with the smaller exponent. Remember that when factoring, you subtract exponents. For instance, in 3x!5)2 ( 2x!3)2 the smaller exponent is ! 52 and the common factor is x!5)2. 3x!5)2 ( 2x!3)2 # x!5)2+3(1' ( 2x!3)2! (!5)2'* # x!5)2(3 ( 2x1' # Example 9 3 ( 2x x 5)2 Simplifying an Expression Simplify the following expression containing negative exponents. x(1 ! 2x'!3)2 ( (1 ! 2x'!1)2 Solution Begin by factoring out the common factor with the smaller exponent. x(1 ! 2x'!3)2 ( (1 ! 2x'!1)2 # (1 ! 2x'!3)2+ x ( (1 ! 2x'(!1)2)!(!3)2)* # (1 ! 2x'!3)2+x ( (1 ! 2x'1* # 1!x (1 ! 2x' 3)2 Now try Exercise 81. A second method for simplifying an expression with negative exponents is shown in the next example. Example 10 Simplifying an Expression with Negative Exponents (4 ! x 2'1)2 ( x 2(4 ! x 2'!1)2 4 ! x2 # (4 ! x 2'1)2 ( x 2(4 ! x 2'!1)2 (4 ! x 2'1)2 * (4 ! x 2'1)2 4 ! x2 # (4 ! x 2'1 ( x 2(4 ! x 2' 0 (4 ! x 2' 3)2 # 4 ! x2 ( x2 (4 ! x 2' 3)2 # 4 (4 ! x 2' 3)2 Now try Exercise 83. Appendix A.4 Example 11 Rational Expressions A45 Rewriting a Difference Quotient The following expression from calculus is an example of a difference quotient. "x ( h ! "x h Rewrite this expression by rationalizing its numerator. Solution "x ( h ! "x h You can review the techniques for rationalizing a numerator in Appendix A.2. # "x ( h ! "x h "x ( h ( "x * "x ( h ( "x # ("x ( h '2 ! ("x '2 h("x ( h ( "x ' # h h("x ( h ( "x ' # 1 , "x ( h ( "x h$0 Notice that the original expression is undefined when h # 0. So, you must exclude h # 0 from the domain of the simplified expression so that the expressions are equivalent. Now try Exercise 89. Difference quotients, such as that in Example 11, occur frequently in calculus. Often, they need to be rewritten in an equivalent form that can be evaluated when h # 0. Note that the equivalent form is not simpler than the original form, but it has the advantage that it is defined when h # 0. A.4 EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises. VOCABULARY: Fill in the blanks. 1. The set of real numbers for which an algebraic expression is defined is the ________ of the expression. 2. The quotient of two algebraic expressions is a fractional expression and the quotient of two polynomials is a ________ ________. 3. Fractional expressions with separate fractions in the numerator, denominator, or both are called ________ fractions. 4. To simplify an expression with negative exponents, it is possible to begin by factoring out the common factor with the ________ exponent. 5. Two algebraic expressions that have the same domain and yield the same values for all numbers in their domains are called ________. 6. An important rational expression, such as a ________ ________. (x ( h'2 ! x2 , that occurs in calculus is called h A46 Appendix A Review of Fundamental Concepts of Algebra 44. Error Analysis SKILLS AND APPLICATIONS In Exercises 7–22, find the domain of the expression. 7. 3x 2 ! 4x ( 7 9. 4x 3 ( 3, x & 0 1 11. 3!x 13. x2 !1 x2 ! 2x ( 1 8. 2x 2 ( 5x ! 2 10. 6x 2 ! 9, x > 0 12. 14. x(6 3x ( 2 x2 ! 5x ( 6 x2 ! 4 x2 ! 2x ! 3 15. 2 x ! 6x ( 9 x2 ! x ! 12 16. 2 x ! 8x ( 16 17. "x ( 7 19. "2x ! 5 18. "4 ! x 20. "4x ( 5 21. 1 22. "x ! 3 x2 1 "x ( 2 15x 2 10x 3xy 27. xy ( x 25. 26. 28. 18y 2 60y 5 45. 30. 9x 2 ( 9x 2x ( 2 31. x!5 10 ! 2x 32. 12 ! 4x x!3 y 2 ! 16 33. y(4 35. 37. 39. 41. 34. x 3 ( 5x 2 ( 6x x2 ! 4 2 y ! 7y ( 12 y 2 ( 3y ! 18 2 ! x ( 2x 2 ! x 3 x2 ! 4 z3 ! 8 2 z ( 2z ( 4 43. Error Analysis 5x3 2x3 ( 4 # 36. 38. 40. 42. x 2 ! 25 5!x x 2 ( 8x ! 20 x 2 ( 11x ( 10 x 2 ! 7x ( 6 x 2 ( 11x ( 10 x2 ! 9 3 x ( x 2 ! 9x ! 9 y 3 ! 2y 2 ! 3y y3 ( 1 Describe the error. 5x3 5 5 # # 3 2x ( 4 2 ( 4 6 1 2 3 4 5 6 x(1 46. 0 x 1 2 3 4 5 6 x!3 x2 ! x ! 6 1 x(2 GEOMETRY In Exercises 47 and 48, find the ratio of the area of the shaded portion of the figure to the total area of the figure. 47. 48. x+5 2 r xy ! y 4y ! 8y 2 10y ! 5 0 x x2 ! 2x ! 3 x!3 2x 2y 29. x(x ( 5'(x ! 5' x(x ( 5' # (x ! 5'(x ( 3' x(3 In Exercises 45 and 46, complete the table. What can you conclude? 3 3( ! ' 24. # 4 4(x ( 1' In Exercises 25–42, write the rational expression in simplest form. x 3 ( 25x x(x 2 ( 25' # ! 2x ! 15 (x ! 5'(x ( 3' # In Exercises 23 and 24, find the missing factor in the numerator such that the two fractions are equivalent. 5 5(!' 23. # 2x 6x 2 Describe the error. 2x + 3 x+5 2 x+5 In Exercises 49–56, perform the multiplication or division and simplify. 5 x!1 x ( 13 x(x ! 3' 50. 3 * * x ! 1 25(x ! 2' x (3 ! x' 5 2 r r 4y ! 16 4!y 51. ) 52. ) r ! 1 r2 ! 1 5y ( 15 2y ( 6 t2 ! t ! 6 t(3 53. 2 * t ( 6t ( 9 t 2 ! 4 x 2 ( xy ! 2y 2 x 54. * 3 2 2 x (x y x ( 3xy ( 2y 2 49. 55. x 2 ! 36 x 3 ! 6x 2 ) 2 x x (x 56. x 2 ! 14x ( 49 3x ! 21 ) x 2 ! 49 x(7 Appendix A.4 In Exercises 57–68, perform the addition or subtraction and simplify. 57. 6 ! 5 x(3 5 x ( x!1 x!1 3 5 61. ( x!2 2!x 59. 63. 4 x ! 2x ( 1 x ( 2 64. 3x1)3 ! x!2)3 3x!2)3 !x 3(1 ! x 2'!1)2 ! 2x(1 ! x 2'1)2 84. x4 83. 1 x ! x 2 ! x ! 2 x 2 ! 5x ( 6 2 10 66. 2 ( 2 x ! x ! 2 x ( 2x ! 8 1 2 1 67. ! ( 2 ( 3 x x (1 x (x 2 2 1 68. ( ( x ( 1 x ! 1 x2 ! 1 x ( 4 3x ! 8 x ( 4 ! 3x ! 8 ! # x(2 x(2 x(2 !2x ! 4 !2(x ( 2' # # # !2 x(2 x(2 6!x x(2 8 70. ( ( 2 2 x(x ( 2' x x (x ( 2' x(6 ! x' ( (x ( 2' 2 ( 8 # x 2(x ( 2' 6x ! x 2 ( x 2 ( 4 ( 8 # x 2(x ( 2' 6(x ( 2' 6 # 2 # x (x ( 2' x 2 69. In Exercises 71–76, simplify the complex fraction. x 73. (x ! 2' x2 (x ( 1'2 1 2 x 1 (x ( 1' 2 3 75. - "x ! (x ! 4' x 4 ! 4 x 2 x !1 x 74. (x ! 1'2 x t2 ! "t 2 ( 1 "t 2 ( 1 76. t2 72. 1 2"x "x . In Exercises 85– 88, simplify the difference quotient. 1 1 1 1 ! ! (x ( h' 2 x 2 x(h x 85. 86. h h x(h x 1 1 ! ! x(h(1 x(1 x(h!4 x!4 87. 88. h h - ERROR ANALYSIS In Exercises 69 and 70, describe the error. 71. x 5 ! 2x!2 78. x5 ! 5x!3 x 2(x 2 ( 1'!5 ! (x 2 ( 1'!4 2x(x ! 5'!3 ! 4x 2(x ! 5'!4 2x 2(x ! 1'1)2 ! 5(x ! 1'!1)2 4x 3(2x ! 1'3)2 ! 2x(2x ! 1'!1)2 In Exercises 83 and 84, simplify the expression. 2 5x ( x ! 3 3x ( 4 65. - 2 ! 1. In Exercises 77–82, factor the expression by removing the common factor with the smaller exponent. 77. 79. 80. 81. 82. 3 !5 x!1 2x ! 1 1 ! x 60. ( x(3 x(3 2x 5 62. ! x!5 5!x 58. - 1 - A47 Rational Expressions . . 2 . 1 . - 2 - . . In Exercises 89–94, simplify the difference quotient by rationalizing the numerator. 89. 91. 93. 94. "x ( 2 ! "x 90. 2 "t ( 3 ! "3 92. t "z ! 3 ! "z 3 "x ( 5 ! "5 x "x ( h ( 1 ! "x ( 1 h "x ( h ! 2 ! "x ! 2 h PROBABILITY In Exercises 95 and 96, consider an experiment in which a marble is tossed into a box whose base is shown in the figure. The probability that the marble will come to rest in the shaded portion of the box is equal to the ratio of the shaded area to the total area of the figure. Find the probability. 95. 96. x 2x + 1 x 2 x x+4 x x+2 4 x (x + 2) 97. RATE A digital copier copies in color at a rate of 50 pages per minute. (a) Find the time required to copy one page. A48 Appendix A Review of Fundamental Concepts of Algebra (b) Find the time required to copy x pages. (c) Find the time required to copy 120 pages. 98. RATE After working together for t hours on a common task, two workers have done fractional parts of the job equal to t)3 and t)5, respectively. What fractional part of the task has been completed? 102. INTERACTIVE MONEY MANAGEMENT The table shows the projected numbers of U.S. households (in millions) banking online and paying bills online from 2002 through 2007. (Source: eMarketer; Forrester Research) FINANCE In Exercises 99 and 100, the formula that approximates the annual interest rate r of a monthly installment loan is given by 24(NM ! P' [ ] N r" 3P # NM 12 4 Year Banking Paying Bills 2002 2003 2004 2005 2006 2007 21.9 26.8 31.5 35.0 40.0 45.0 13.7 17.4 20.9 23.9 26.7 29.1 where N is the total number of payments, M is the monthly payment, and P is the amount financed. Mathematical models for these data are 99. (a) Approximate the annual interest rate for a four-year car loan of $20,000 that has monthly payments of $475. (b) Simplify the expression for the annual interest rate r, and then rework part (a). 100. (a) Approximate the annual interest rate for a fiveyear car loan of $28,000 that has monthly payments of $525. (b) Simplify the expression for the annual interest rate r, and then rework part (a). 101. REFRIGERATION When food (at room temperature) is placed in a refrigerator, the time required for the food to cool depends on the amount of food, the air circulation in the refrigerator, the original temperature of the food, and the temperature of the refrigerator. The model that gives the temperature of food that has an original temperature of 75,F and is placed in a 40,F refrigerator is Number banking online # T # 10 -t 4t 2 ( 16t ( 75 2 ( 4t ( 10 . 0 2 4 6 8 10 12 T t 14 16 18 and Number paying bills online # 4.39t ( 5.5 0.002t2 ( 0.01t ( 1.0 where t represents the year, with t # 2 corresponding to 2002. (a) Using the models, create a table to estimate the projected numbers of households banking online and the projected numbers of households paying bills online for the given years. (b) Compare the values given by the models with the actual data. (c) Determine a model for the ratio of the projected number of households paying bills online to the projected number of households banking online. (d) Use the model from part (c) to find the ratios for the given years. Interpret your results. EXPLORATION where T is the temperature (in degrees Fahrenheit) and t is the time (in hours). (a) Complete the table. t !0.728t2 ( 23.81t ! 0.3 !0.049t2 ( 0.61t ( 1.0 20 22 T (b) What value of T does the mathematical model appear to be approaching? TRUE OR FALSE? In Exercises 103 and 104, determine whether the statement is true or false. Justify your answer. 103. x 2n ! 12n # x n ( 1n x n ! 1n 104. x 2 ! 3x ( 2 # x ! 2, for all values of x x!1 105. THINK ABOUT IT How do you determine whether a rational expression is in simplest form? 106. CAPSTONE In your own words, explain how to divide rational expressions. Appendix A.5 Solving Equations A49 A.5 SOLVING EQUATIONS What you should learn • Identify different types of equations. • Solve linear equations in one variable and equations that lead to linear equations. • Solve quadratic equations by factoring, extracting square roots, completing the square, and using the Quadratic Formula. • Solve polynomial equations of degree three or greater. • Solve equations involving radicals. • Solve equations with absolute values. • Use common formulas to solve reallife problems. Why you should learn it Linear equations are used in many real-life applications. For example, in Exercises 155 and 156 on pages A61 and A62, linear equations can be used to model the relationship between the length of a thigh bone and the height of a person, helping researchers learn about ancient cultures. Equations and Solutions of Equations An equation in x is a statement that two algebraic expressions are equal. For example 3x ! 5 # 7, x 2 ! x ! 6 # 0, and "2x # 4 are equations. To solve an equation in x means to find all values of x for which the equation is true. Such values are solutions. For instance, x # 4 is a solution of the equation 3x ! 5 # 7 because 3(4' ! 5 # 7 is a true statement. The solutions of an equation depend on the kinds of numbers being considered. For instance, in the set of rational numbers, x 2 # 10 has no solution because there is no rational number whose square is 10. However, in the set of real numbers, the equation has the two solutions x # "10 and x # ! "10. An equation that is true for every real number in the domain of the variable is called an identity. The domain is the set of all real numbers for which the equation is defined. For example x2 ! 9 # (x ( 3'(x ! 3' Identity is an identity because it is a true statement for any real value of x. The equation x 1 # 3x2 3x Identity where x $ 0, is an identity because it is true for any nonzero real value of x. An equation that is true for just some (or even none) of the real numbers in the domain of the variable is called a conditional equation. For example, the equation x2 ! 9 # 0 Conditional equation is conditional because x # 3 and x # !3 are the only values in the domain that satisfy the equation. The equation 2x ! 4 # 2x ( 1 is conditional because there are no real values of x for which the equation is true. Linear Equations in One Variable Definition of a Linear Equation A linear equation in one variable x is an equation that can be written in the standard form ax ( b # 0 where a and b are real numbers with a $ 0. A50 Appendix A Review of Fundamental Concepts of Algebra A linear equation in one variable, written in standard form, always has exactly one solution. To see this, consider the following steps. Note that some linear equations in nonstandard form have no solution or infinitely many solutions. For instance, x#x(1 has no solution because it is not true for any value of x. Because 5x ( 10 # 5(x ( 2' is true for any value of x, the equation has infinitely many solutions. ax ( b # 0 Original equation, with a $ 0 ax # !b x#! b a Subtract b from each side. Divide each side by a. To solve a conditional equation in x, isolate x on one side of the equation by a sequence of equivalent (and usually simpler) equations, each having the same solution(s) as the original equation. The operations that yield equivalent equations come from the Substitution Principle (see Appendix A.1) and simplification techniques. Generating Equivalent Equations An equation can be transformed into an equivalent equation by one or more of the following steps. Given Equation 2x ! x # 4 Equivalent Equation x#4 2. Add (or subtract) the same quantity to (from) each side of the equation. x(1#6 x#5 3. Multiply (or divide) each side of the equation by the same nonzero quantity. 2x # 6 x#3 4. Interchange the two sides of the equation. 2#x x#2 1. Remove symbols of grouping, combine like terms, or simplify fractions on one or both sides of the equation. After solving an equation, you should check each solution in the original equation. For instance, you can check the solution of Example 1(a) as follows. 3x ! 6 # 0 ? 3(2' ! 6 # 0 0#0 Write original equation. Try checking the solution of Example 1(b). Solving a Linear Equation a. 3x ! 6 # 0 Original equation 3x # 6 Add 6 to each side. x#2 Divide each side by 3. b. 5x ( 4 # 3x ! 8 Substitute 2 for x. Solution checks. Example 1 ✓ 2x ( 4 # !8 2x # !12 x # !6 Original equation Subtract 3x from each side. Subtract 4 from each side. Divide each side by 2. Now try Exercise 15. Appendix A.5 An equation with a single fraction on each side can be cleared of denominators by cross multiplying. To do this, multiply the left numerator by the right denominator and the right numerator by the left denominator as follows. a c # b d ad # cb A51 To solve an equation involving fractional expressions, find the least common denominator (LCD) of all terms and multiply every term by the LCD. This process will clear the original equation of fractions and produce a simpler equation. Example 2 Solve An Equation Involving Fractional Expressions x 3x ( # 2. 3 4 Solution 3x x ( #2 3 4 Original equation Cross multiply. Solving Equations Write original equation. x 3x (12' ( (12' # (12'2 3 4 Multiply each term by the LCD of 12. 4x ( 9x # 24 Divide out and multiply. 13x # 24 x# Combine like terms. 24 13 Divide each side by 13. The solution is x # 24 13 . Check this in the original equation. Now try Exercise 23. When multiplying or dividing an equation by a variable quantity, it is possible to introduce an extraneous solution. An extraneous solution is one that does not satisfy the original equation. Therefore, it is essential that you check your solutions. Example 3 Solve An Equation with an Extraneous Solution 1 3 6x . # ! x ! 2 x ( 2 x2 ! 4 Solution The LCD is x 2 ! 4, or (x ( 2'(x ! 2'. Multiply each term by this LCD. Recall that the least common denominator of two or more fractions consists of the product of all prime factors in the denominators, with each factor given the highest power of its occurrence in any denominator. For instance, in Example 3, by factoring each denominator you can determine that the LCD is (x ( 2'(x ! 2'. 1 3 6x (x ( 2'(x ! 2' # (x ( 2'(x ! 2' ! 2 (x ( 2'(x ! 2' x!2 x(2 x !4 x ( 2 # 3(x ! 2' ! 6x, x $ ±2 x ( 2 # 3x ! 6 ! 6x x ( 2 # !3x ! 6 4x # !8 x # !2 Extraneous solution In the original equation, x # !2 yields a denominator of zero. So, x # !2 is an extraneous solution, and the original equation has no solution. Now try Exercise 35. A52 Appendix A Review of Fundamental Concepts of Algebra Quadratic Equations A quadratic equation in x is an equation that can be written in the general form ax2 ( bx ( c # 0 where a, b, and c are real numbers, with a $ 0. A quadratic equation in x is also known as a second-degree polynomial equation in x. You should be familiar with the following four methods of solving quadratic equations. Solving a Quadratic Equation Factoring: If ab # 0, then a # 0 or b # 0. x2 ! x ! 6 # 0 Example: (x ! 3'(x ( 2' # 0 x!3#0 x#3 x(2#0 x # !2 Square Root Principle: If u 2 # c, where c > 0, then u # ± "c. (x ( 3'2 # 16 Example: The Square Root Principle is also referred to as extracting square roots. x ( 3 # ±4 x # !3 ± 4 or x#1 x # !7 Completing the Square: If x 2 ( bx # c, then -2. 2 -x ( 2 . 2 x 2 ( bx ( b b Example: #c( -2. #c( b2 . 4 b 2 Add -b2. 2 Add -62. 2 to each side. x 2 ( 6x # 5 x 2 ( 6x ( 32 # 5 ( 32 to each side. (x ( 3' # 14 2 x ( 3 # ± "14 x # !3 ± "14 Quadratic Formula: If ax 2 ( bx ( c # 0, then x # You can solve every quadratic equation by completing the square or using the Quadratic Formula. Example: !b ± "b2 ! 4ac . 2a 2x 2 ( 3x ! 1 # 0 x# # !3 ± "32 ! 4(2'(!1' 2(2' !3 ± "17 4 Appendix A.5 Example 4 a. Solving Equations A53 Solving a Quadratic Equation by Factoring 2x 2 ( 9x ( 7 # 3 Original equation 2x2 Write in general form. ( 9x ( 4 # 0 (2x ( 1'(x ( 4' # 0 Factor. 1 2 2x ( 1 # 0 x#! x(4#0 x # !4 Set 1st factor equal to 0. Set 2nd factor equal to 0. The solutions are x # ! 12 and x # !4. Check these in the original equation. b. 6x 2 ! 3x # 0 Original equation 3x(2x ! 1' # 0 3x # 0 2x ! 1 # 0 Factor. x#0 x# 1 2 Set 1st factor equal to 0. Set 2nd factor equal to 0. The solutions are x # 0 and x # 12. Check these in the original equation. Now try Exercise 49. Note that the method of solution in Example 4 is based on the Zero-Factor Property from Appendix A.1. Be sure you see that this property works only for equations written in general form (in which the right side of the equation is zero). So, all terms must be collected on one side before factoring. For instance, in the equation (x ! 5'(x ( 2' # 8, it is incorrect to set each factor equal to 8. Try to solve this equation correctly. Example 5 Extracting Square Roots Solve each equation by extracting square roots. b. (x ! 3'2 # 7 a. 4x 2 # 12 Solution a. 4x 2 # 12 x2 Write original equation. #3 Divide each side by 4. x # ± "3 Extract square roots. When you take the square root of a variable expression, you must account for both positive and negative solutions. So, the solutions are x # "3 and x # ! "3. Check these in the original equation. b. (x ! 3'2 # 7 x ! 3 # ± "7 x # 3 ± "7 Write original equation. Extract square roots. Add 3 to each side. The solutions are x # 3 ± "7. Check these in the original equation. Now try Exercise 65. A54 Appendix A Review of Fundamental Concepts of Algebra When solving quadratic equations by completing the square, you must add (b)2' 2 to each side in order to maintain equality. If the leading coefficient is not 1, you must divide each side of the equation by the leading coefficient before completing the square, as shown in Example 7. Example 6 Completing the Square: Leading Coefficient Is 1 Solve x 2 ( 2x ! 6 # 0 by completing the square. Solution x 2 ( 2x ! 6 # 0 Write original equation. x 2 ( 2x # 6 2 Add 6 to each side. 2 2 x ( 2x ( 1 # 6 ( 1 Add 12 to each side. 2 (half of 2' (x ( 1'2 # 7 Simplify. x ( 1 # ± "7 x # !1 ± "7 Take square root of each side. Subtract 1 from each side. The solutions are x # !1 ± "7. Check these in the original equation. Now try Exercise 73. Example 7 Completing the Square: Leading Coefficient Is Not 1 3x2 ! 4x ! 5 # 0 Original equation 3x2 ! 4x # 5 Add 5 to each side. 4 5 x2 ! x # 3 3 - . 4 2 x2 ! x ( ! 3 3 2 # Divide each side by 3. - . 5 2 ( ! 3 3 2 Add (! 3 ' to each side. 2 2 (half of ! 43 '2 4 4 19 x2 ! x ( # 3 9 9 - x! 2 3 x! . 2 # 19 9 "19 2 # ± 3 3 x# "19 2 ± 3 3 Now try Exercise 77. Simplify. Perfect square trinomial Extract square roots. Solutions Appendix A.5 WARNING / CAUTION When using the Quadratic Formula, remember that before the formula can be applied, you must first write the quadratic equation in general form. Example 8 Solving Equations A55 The Quadratic Formula: Two Distinct Solutions Use the Quadratic Formula to solve x 2 ( 3x # 9. Solution x2 ( 3x # 9 Write original equation. 2 x ( 3x ! 9 # 0 Write in general form. x# !b ± "b2 ! 4ac 2a Quadratic Formula x# !3 ± "(3'2 ! 4(1'(!9' 2(1' Substitute a # 1, b # 3, and c # !9. x# !3 ± "45 2 Simplify. x# !3 ± 3"5 2 Simplify. The equation has two solutions: x# !3 ( 3"5 2 and x# !3 ! 3"5 . 2 Check these in the original equation. Now try Exercise 87. Example 9 The Quadratic Formula: One Solution Use the Quadratic Formula to solve 8x 2 ! 24x ( 18 # 0. Solution 8x2 ! 24x ( 18 # 0 4x2 ! 12x ( 9 # 0 Write original equation. Divide out common factor of 2. x# !b ± "b2 ! 4ac 2a Quadratic Formula x# ! (!12' ± "(!12'2 ! 4(4'(9' 2(4' Substitute a # 4, b # !12, and c # 9. x# 12 ± "0 3 # 8 2 Simplify. This quadratic equation has only one solution: x # 32. Check this in the original equation. Now try Exercise 91. Note that Example 9 could have been solved without first dividing out a common factor of 2. Substituting a # 8, b # !24, and c # 18 into the Quadratic Formula produces the same result. A56 Appendix A Review of Fundamental Concepts of Algebra Polynomial Equations of Higher Degree WARNING / CAUTION A common mistake that is made in solving equations such as the equation in Example 10 is to divide each side of the equation by the variable factor x 2. This loses the solution x # 0. When solving an equation, always write the equation in general form, then factor the equation and set each factor equal to zero. Do not divide each side of an equation by a variable factor in an attempt to simplify the equation. The methods used to solve quadratic equations can sometimes be extended to solve polynomial equations of higher degree. Example 10 Solving a Polynomial Equation by Factoring Solve 3x 4 # 48x 2. Solution First write the polynomial equation in general form with zero on one side, factor the other side, and then set each factor equal to zero and solve. 3x 4 # 48x 2 3x 4 ! 48x 2 Write original equation. #0 Write in general form. 3x 2(x 2 ! 16' # 0 Factor out common factor. 3x 2(x ( 4'(x ! 4' # 0 3x 2 # 0 Write in factored form. x#0 Set 1st factor equal to 0. x(4#0 x # !4 Set 2nd factor equal to 0. x!4#0 x#4 Set 3rd factor equal to 0. You can check these solutions by substituting in the original equation, as follows. Check 3(0'4 # 48(0' 2 3(!4' # 48(!4' 4 0 checks. 2 ✓ !4 checks. 3(4'4 # 48(4' 2 4 checks. ✓ ✓ So, you can conclude that the solutions are x # 0, x # !4, and x # 4. Now try Exercise 113. Example 11 Solving a Polynomial Equation by Factoring Solve x 3 ! 3x 2 ! 3x ( 9 # 0. Solution x3 ! 3x 2 ! 3x ( 9 # 0 Write original equation. x2(x ! 3' ! 3(x ! 3' # 0 Factor by grouping. (x ! 3'(x 2 ! 3' # 0 x!3#0 x2 ! 3 # 0 Distributive Property x#3 Set 1st factor equal to 0. x # ± "3 Set 2nd factor equal to 0. The solutions are x # 3, x # "3, and x # ! "3. Check these in the original equation. Now try Exercise 119. Appendix A.5 Solving Equations A57 Equations Involving Radicals Operations such as squaring each side of an equation, raising each side of an equation to a rational power, and multiplying each side of an equation by a variable quantity all can introduce extraneous solutions. So, when you use any of these operations, checking your solutions is crucial. Example 12 Solving Equations Involving Radicals a. "2x ( 7 ! x # 2 Original equation "2x ( 7 # x ( 2 2x ( 7 # x2 Isolate radical. ( 4x ( 4 Square each side. 0 # x 2 ( 2x ! 3 Write in general form. 0 # (x ( 3'(x ! 1' Factor. x(3#0 x # !3 Set 1st factor equal to 0. x!1#0 x#1 Set 2nd factor equal to 0. By checking these values, you can determine that the only solution is x # 1. b. "2x ! 5 ! "x ! 3 # 1 Original equation "2x ! 5 # "x ! 3 ( 1 When an equation contains two radicals, it may not be possible to isolate both. In such cases, you may have to raise each side of the equation to a power at two different stages in the solution, as shown in Example 12(b). Isolate "2x ! 5. 2x ! 5 # x ! 3 ( 2"x ! 3 ( 1 Square each side. 2x ! 5 # x ! 2 ( 2"x ! 3 Combine like terms. x ! 3 # 2"x ! 3 Isolate 2"x ! 3. x 2 ! 6x ( 9 # 4(x ! 3' Square each side. x 2 ! 10x ( 21 # 0 Write in general form. (x ! 3'(x ! 7' # 0 Factor. x!3#0 x#3 Set 1st factor equal to 0. x!7#0 x#7 Set 2nd factor equal to 0. The solutions are x # 3 and x # 7. Check these in the original equation. Now try Exercise 129. Example 13 Solving an Equation Involving a Rational Exponent (x ! 4'2)3 # 25 3 " (x ! 4'2 # 25 (x ! 4' # 15,625 2 x ! 4 # ± 125 x # 129, x # !121 Now try Exercise 137. Original equation Rewrite in radical form. Cube each side. Take square root of each side. Add 4 to each side. A58 Appendix A Review of Fundamental Concepts of Algebra Equations with Absolute Values To solve an equation involving an absolute value, remember that the expression inside the absolute value signs can be positive or negative. This results in two separate equations, each of which must be solved. For instance, the equation ,x ! 2, # 3 results in the two equations x ! 2 # 3 and ! (x ! 2' # 3, which implies that the equation has two solutions: x # 5 and x # !1. Example 14 , Solving an Equation Involving Absolute Value , Solve x 2 ! 3x # !4x ( 6. Solution Because the variable expression inside the absolute value signs can be positive or negative, you must solve the following two equations. First Equation x 2 ! 3x # !4x ( 6 Use positive expression. 2 x (x!6#0 Write in general form. (x ( 3'(x ! 2' # 0 Factor. x(3#0 x # !3 Set 1st factor equal to 0. x!2#0 x#2 Set 2nd factor equal to 0. Second Equation ! (x 2 ! 3x' # !4x ( 6 Use negative expression. x 2 ! 7x ( 6 # 0 Write in general form. (x ! 1'(x ! 6' # 0 Factor. x!1#0 x#1 Set 1st factor equal to 0. x!6#0 x#6 Set 2nd factor equal to 0. Check ? ,(!3'2 ! 3(!3', # !4(!3' ( 6 Substitute !3 for x. ✓ 18 # 18 ? (2'2 ! 3(2' # !4(2' ( 6 !3 checks. 2 $ !2 ? (1'2 ! 3(1' # !4(1' ( 6 2 does not check. 2#2 ? (6'2 ! 3(6' # !4(6' ( 6 1 checks. , , , , , , 18 $ !18 The solutions are x # !3 and x # 1. Now try Exercise 151. Substitute 2 for x. Substitute 1 for x. ✓ Substitute 6 for x. 6 does not check. Appendix A.5 Solving Equations A59 Common Formulas The following geometric formulas are used at various times throughout this course. For your convenience, some of these formulas along with several others are also provided on the inside cover of this text. Common Formulas for Area A, Perimeter P, Circumference C, and Volume V Rectangle Circle A # lw A # " r2 P # 2l ( 2w C # 2" r w Triangle 1 A # bh 2 P#a(b(c l V # lwh V # " r 2h 4 V # " r3 3 r h w r l h Using a Geometric Formula A cylindrical can has a volume of 200 cubic centimeters (cm3' and a radius of 4 centimeters (cm), as shown in Figure A.13. Find the height of the can. 4 cm h A.13 Sphere b Example 15 FIGURE Circular Cylinder c a h r Rectangular Solid Solution The formula for the volume of a cylinder is V # " r 2h. To find the height of the can, solve for h. h# V "r2 Then, using V # 200 and r # 4, find the height. h# # 200 " (4' 2 Substitute 200 for V and 4 for r. 200 16" Simplify denominator. & 3.98 Use a calculator. Because the value of h was rounded in the solution, a check of the solution will not result in an equality. If the solution is valid, the expressions on each side of the equal sign will be approximately equal to each other. V # "r2 h ? 200 & "(4'2(3.98' Write original equation. 200 & 200.06 Solution checks. Substitute 200 for V, 4 for r, and 3.98 for h. ✓ You can also use unit analysis to check that your answer is reasonable. 200 cm3 & 3.98 cm 16" cm2 Now try Exercise 157. A60 A.5 Appendix A Review of Fundamental Concepts of Algebra EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises. VOCABULARY: Fill in the blanks. 1. A(n) ________ is a statement that equates two algebraic expressions. 2. A linear equation in one variable is an equation that can be written in the standard form ________. 3. When solving an equation, it is possible to introduce an ________ solution, which is a value that does not satisfy the original equation. 4. The four methods that can be used to solve a quadratic equation are ________, ________, ________, and the ________. SKILLS AND APPLICATIONS In Exercises 5–12, determine whether the equation is an identity or a conditional equation. 4(x ( 1' # 4x ( 4 2(x ! 3' # 7x ! 1 !6(x ! 3' ( 5 # !2x ( 10 3(x ( 2' ! 5 # 3x ( 1 4(x ( 1' ! 2x # 2(x ( 2' x 2 ( 2(3x ! 2' # x 2 ( 6x ! 4 1 4x 5 3 11. 3 ( # 12. ( # 24 x(1 x(1 x x 5. 6. 7. 8. 9. 10. In Exercises 13–26, solve the equation and check your solution. 13. 15. 17. 19. 20. 21. 22. x ( 11 # 15 14. 7 ! x # 19 7 ! 2x # 25 16. 7x ( 2 # 23 8x ! 5 # 3x ( 20 18. 7x ( 3 # 3x ! 17 4y ( 2 ! 5y # 7 ! 6y 3(x ( 3' # 5(1 ! x' ! 1 x ! 3(2x ( 3' # 8 ! 5x 9x ! 10 # 5x ( 2(2x ! 5' x x 3x 3x 4x 23. ! #4 24. ! # 3 ( 8 3 5 2 10 3 1 25. 2(z ( 5' ! 4(z ( 24' # 0 26. 0.60x ( 0.40(100 ! x' # 50 In Exercises 27–42, solve the equation and check your solution. (If not possible, explain why.) 27. x ( 8 # 2(x ! 2' ! x 28. 8(x ( 2' ! 3(2x ( 1' # 2(x ( 5' 100 ! 4x 5x ( 6 29. # (6 3 4 17 ( y 32 ( y 30. ( # 100 y y 5x ! 4 2 6 15 31. # 32. !4# (3 5x ( 4 3 x x 1 2 34. ( z(2 x x x 4 ( (2#0 x(4 x(4 8x 7 ! # !4 2x ( 1 2x ! 1 2 1 2 # ( (x ! 4'(x ! 2' x ! 4 x ! 2 1 3 4 ( # x ! 2 x ( 3 x2 ( x ! 6 3 4 1 6 40. ! ( # 2 x ! 3x x x!3 x x (x ( 2'2 ( 5 # (x ( 3'2 (2x ( 1'2 # 4(x 2 ( x ( 1' 33. 3 # 2 ( 35. 36. 37. 38. 39. 41. 42. 2 #0 !5 2 3(x ( 5' # 2 (3 x ( 3x In Exercises 43– 46, write the quadratic equation in general form. 43. 2x 2 # 3 ! 8x 45. 15(3x 2 ! 10' # 18x 44. 13 ! 3(x ( 7'2 # 0 46. x(x ( 2' # 5x 2 ( 1 In Exercises 47– 58, solve the quadratic equation by factoring. 47. 49. 51. 53. 55. 57. 58. 48. 9x 2 ! 4 # 0 6x 2 ( 3x # 0 50. x 2 ! 10x ( 9 # 0 x 2 ! 2x ! 8 # 0 52. 4x 2 ( 12x ( 9 # 0 x2 ! 12x ( 35 # 0 2 54. 2x 2 # 19x ( 33 3 ( 5x ! 2x # 0 56. 18 x 2 ! x ! 16 # 0 x 2 ( 4x # 12 x 2 ( 2ax ( a 2 # 0, a is a real number (x ( a'2 ! b 2 # 0, a and b are real numbers In Exercises 59–70, solve the equation by extracting square roots. 59. 61. 63. 65. 67. 69. x 2 # 49 3x 2 # 81 (x ! 12'2 # 16 (x ( 2' 2 # 14 (2x ! 1'2 # 18 (x ! 7'2 # (x ( 3' 2 60. 62. 64. 66. 68. 70. x 2 # 32 9x 2 # 36 (x ( 13'2 # 25 (x ! 5'2 # 30 (2x ( 3'2 ! 27 # 0 (x ( 5'2 # (x ( 4' 2 Appendix A.5 In Exercises 71– 80, solve the quadratic equation by completing the square. 71. 73. 75. 77. 79. x 2 ( 4x ! 32 # 0 x 2 ( 12x ( 25 # 0 8 ( 4x ! x 2 # 0 2x 2 ( 5x ! 8 # 0 5x2 ! 15x ( 7 # 0 72. 74. 76. 78. 80. x2 ( 6x ( 2 # 0 x 2 ( 8x ( 14 # 0 9x 2 ! 12x # 14 4x 2 ! 4x ! 99 # 0 3x2 ( 9x ( 5 # 0 In Exercises 81–98, use the Quadratic Formula to solve the equation. 81. 83. 85. 87. 89. 91. 93. 95. 97. 2x 2 ( x ! 1 # 0 2 ( 2x ! x 2 # 0 x 2 ( 14x ( 44 # 0 x 2 ( 8x ! 4 # 0 12x ! 9x 2 # !3 9x2 ( 24x ( 16 # 0 28x ! 49x 2 # 4 8t # 5 ( 2t 2 ( y ! 5'2 # 2y 82. 84. 86. 88. 90. 92. 94. 96. 98. 25x 2 ! 20x ( 3 # 0 x 2 ! 10x ( 22 # 0 6x # 4 ! x 2 4x 2 ! 4x ! 4 # 0 16x 2 ( 22 # 40x 16x 2 ! 40x ( 5 # 0 3x ( x 2 ! 1 # 0 25h2 ( 80h ( 61 # 0 (57x ! 14'2 # 8x In Exercises 99–104, use the Quadratic Formula to solve the equation. (Round your answer to three decimal places.) 99. 100. 101. 102. 103. 104. 0.1x2 ( 0.2x ! 0.5 # 0 2x 2 ! 2.50x ! 0.42 # 0 !0.067x 2 ! 0.852x ( 1.277 # 0 !0.005x 2 ( 0.101x ! 0.193 # 0 422x 2 ! 506x ! 347 # 0 !3.22x 2 ! 0.08x ( 28.651 # 0 In Exercises 105–112, solve the equation using any convenient method. 105. 107. 109. 111. 112. ! 2x ! 1 # 0 106. ( 33x # 0 2 2 (x ( 3' # 81 108. x ! 14x ( 49 # 0 11 2 x !x! 4 #0 110. 3x ( 4 # 2x2 ! 7 4x 2 ( 2x ( 4 # 2x ( 8 a 2x 2 ! b 2 # 0, a and b are real numbers, a $ 0 x2 11x 2 In Exercises 113–126, find all real solutions of the equation. Check your solutions in the original equation. 113. 115. 117. 118. 119. 120. 2x4 ! 50x2 # 0 114. 20x3 ! 125x # 0 x 4 ! 81 # 0 116. x6 ! 64 # 0 3 x ( 216 # 0 9x4 ! 24x3 ( 16x 2 # 0 x3 ! 3x2 ! x ( 3 # 0 x3 ( 2x2 ( 3x ( 6 # 0 121. 122. 123. 125. x4 x4 x4 x6 ( ! ! ( Solving Equations A61 x # x3 ( 1 2x3 # 16 ( 8x ! 4x3 124. 36t 4 ( 29t 2 ! 7 # 0 4x2 ( 3 # 0 126. x6 ( 3x3 ( 2 # 0 7x3 ! 8 # 0 In Exercises 127–154, find all solutions of the equation. Check your solutions in the original equation. 127. 129. 131. 133. 135. 137. 139. 141. 142. 143. 145. 147. 149. 151. 153. 128. 130. 132. 134. 136. 138. 140. "2x ! 10 # 0 7"x ! 6 # 0 "x ! 10 ! 4 # 0 "5 ! x ! 3 # 0 "2x ( 5 ( 3 # 0 "3 ! 2x ! 2 # 0 3 3 " " 2x ( 1 ( 8 # 0 4x ! 3 ( 2 # 0 "5x ! 26 ( 4 # x "x ( 5 # "2x ! 5 (x ! 6'3)2 # 8 (x ( 3'3)2 # 8 (x ( 3'2)3 # 5 (x2 ! x ! 22'4)3 # 16 3x(x ! 1'1)2 ( 2(x ! 1'3)2 # 0 4x2(x ! 1'1)3 ( 6x(x ! 1'4)3 # 0 3 1 4 3 144. x# ( ! #1 x 2 x(1 x(2 20 ! x 3 146. 4x ( 1 # #x x x x 1 x(1 x(1 ( # 3 148. ! #0 x2 ! 4 x ( 2 3 x(2 150. 13x ( 1 # 12 2x ! 1 # 5 2 152. x 2 ( 6x # 3x ( 18 x #x (x!3 154. x ! 10 # x 2 ! 10x x ( 1 # x2 ! 5 , , ,, , , , , , , , , ANTHROPOLOGY In Exercises 155 and 156, use the following information. The relationship between the length of an adult’s femur (thigh bone) and the height of the adult can be approximated by the linear equations y " 0.432 x ! 10.44 Female y " 0.449x ! 12.15 Male where y is the length of the femur in inches and x is the height of the adult in inches (see figure). x in. y in. femur 155. An anthropologist discovers a femur belonging to an adult human female. The bone is 16 inches long. Estimate the height of the female. A62 Appendix A Review of Fundamental Concepts of Algebra 156. From the foot bones of an adult human male, an anthropologist estimates that the person’s height was 69 inches. A few feet away from the site where the foot bones were discovered, the anthropologist discovers a male adult femur that is 19 inches long. Is it likely that both the foot bones and the thigh bone came from the same person? 157. VOLUME OF A BILLIARD BALL A billiard ball has a volume of 5.96 cubic inches. Find the radius of a billiard ball. 158. LENGTH OF A TANK The diameter of a cylindrical propane gas tank is 4 feet. The total volume of the tank is 603.2 cubic feet. Find the length of the tank. 159. GEOMETRY A “Slow Moving Vehicle” sign has the shape of an equilateral triangle. The sign has a perimeter of 129 centimeters. Find the length of each side of the sign. Find the area of the sign. 160. GEOMETRY The radius of a traffic cone is 14 centimeters and the lateral surface area of the cone is 1617 square centimeters. Find the height of the cone. 161. VOTING POPULATION The total voting-age population P (in millions) in the United States from 1990 through 2006 can be modeled by P# 182.17 ! 1.542t , 0 % t % 16 1 ! 0.018t where t represents the year, with t # 0 corresponding to 1990. (Source: U.S. Census Bureau) (a) In which year did the total voting-age population reach 200 million? (b) Use the model to predict the year in which the total voting-age population will reach 241 million. Is this prediction reasonable? Explain. 162. AIRLINE PASSENGERS An airline offers daily flights between Chicago and Denver. The total monthly cost C (in millions of dollars) of these flights is C # "0.2x ( 1, where x is the number of passengers (in thousands). The total cost of the flights for June is 2.5 million dollars. How many passengers flew in June? EXPLORATION TRUE OR FALSE? In Exercises 163 and 164, determine whether the statement is true or false. Justify your answer. 163. An equation can never have more than one extraneous solution. 164. When solving an absolute value equation, you will always have to check more than one solution. 165. THINK ABOUT IT What is meant by equivalent equations? Give an example of two equivalent equations. 166. Solve 3(x ( 4'2 ( (x ( 4' ! 2 # 0 in two ways. (a) Let u # x ( 4, and solve the resulting equation for u. Then solve the u-solution for x. (b) Expand and collect like terms in the equation, and solve the resulting equation for x. (c) Which method is easier? Explain. THINK ABOUT IT In Exercises 167–170, write a quadratic equation that has the given solutions. (There are many correct answers.) 167. 168. 169. 170. !3 and 6 !4 and !11 1 ( "2 and 1 ! "2 !3 ( "5 and !3 ! "5 In Exercises 171 and 172, consider an equation of the form x # x ! a " b, where a and b are constants. , , 171. Find a and b when the solution of the equation is x # 9. (There are many correct answers.) 172. WRITING Write a short paragraph listing the steps required to solve this equation involving absolute values, and explain why it is important to check your solutions. In Exercises 173 and 174, consider an equation of the form x # "x ! a " b, where a and b are constants. 173. Find a and b when the solution of the equation is x # 20. (There are many correct answers.) 174. WRITING Write a short paragraph listing the steps required to solve this equation involving radicals, and explain why it is important to check your solutions. 175. Solve each equation, given that a and b are not zero. (a) ax 2 ( bx # 0 (b) ax 2 ! ax # 0 176. CAPSTONE (a) Explain the difference between a conditional equation and an identity. (b) Give an example of an absolute value equation that has only one solution. (c) State the Quadratic Formula in words. (d) Does raising each side of an equation to the nth power always yield an equivalent equation? Explain. Appendix A.6 Linear Inequalities in One Variable A63 A.6 LINEAR INEQUALITIES IN ONE VARIABLE What you should learn • Represent solutions of linear inequalities in one variable. • Use properties of inequalities to create equivalent inequalities. • Solve linear inequalities in one variable. • Solve inequalities involving absolute values. • Use inequalities to model and solve real-life problems. Why you should learn it Inequalities can be used to model and solve real-life problems. For instance, in Exercise 121 on page A71, you will use a linear inequality to analyze the average salary for elementary school teachers. Introduction Simple inequalities were discussed in Appendix A.1. There, you used the inequality symbols <, ≤, >, and ≥ to compare two numbers and to denote subsets of real numbers. For instance, the simple inequality x & 3 denotes all real numbers x that are greater than or equal to 3. Now, you will expand your work with inequalities to include more involved statements such as 5x ! 7 < 3x ( 9 and !3 % 6x ! 1 < 3. As with an equation, you solve an inequality in the variable x by finding all values of x for which the inequality is true. Such values are solutions and are said to satisfy the inequality. The set of all real numbers that are solutions of an inequality is the solution set of the inequality. For instance, the solution set of x(1 < 4 is all real numbers that are less than 3. The set of all points on the real number line that represents the solution set is the graph of the inequality. Graphs of many types of inequalities consist of intervals on the real number line. See Appendix A.1 to review the nine basic types of intervals on the real number line. Note that each type of interval can be classified as bounded or unbounded. Example 1 Intervals and Inequalities Write an inequality to represent each interval, and state whether the interval is bounded or unbounded. a. (!3, 5* b. (!3, '' c. +0, 2* d. (! ', '' Solution a. (!3, 5* corresponds to !3 < x % 5. b. (!3, '' corresponds to !3 < x. c. +0, 2* corresponds to 0 % x % 2. d. (! ', '' corresponds to ! ' < x < '. Now try Exercise 9. Bounded Unbounded Bounded Unbounded A64 Appendix A Review of Fundamental Concepts of Algebra Properties of Inequalities The procedures for solving linear inequalities in one variable are much like those for solving linear equations. To isolate the variable, you can make use of the Properties of Inequalities. These properties are similar to the properties of equality, but there are two important exceptions. When each side of an inequality is multiplied or divided by a negative number, the direction of the inequality symbol must be reversed. Here is an example. !2 < 5 Original inequality (!3'(!2' > (!3'(5' Multiply each side by !3 and reverse inequality. 6 > !15 Simplify. Notice that if the inequality was not reversed, you would obtain the false statement 6 < !15. Two inequalities that have the same solution set are equivalent. For instance, the inequalities x(2 < 5 and x < 3 are equivalent. To obtain the second inequality from the first, you can subtract 2 from each side of the inequality. The following list describes the operations that can be used to create equivalent inequalities. Properties of Inequalities Let a, b, c, and d be real numbers. 1. Transitive Property a < b and b < c a < c 2. Addition of Inequalities a < b and c < d a(c < b(d 3. Addition of a Constant a < b a(c < b(c 4. Multiplication by a Constant For c > 0, a < b ac < bc For c < 0, a < b ac > bc Reverse the inequality. Each of the properties above is true if the symbol < is replaced by % and the symbol > is replaced by ≥. For instance, another form of the multiplication property would be as follows. For c > 0, a % b ac % bc For c < 0, a % b ac & bc Appendix A.6 Linear Inequalities in One Variable A65 Solving a Linear Inequality in One Variable The simplest type of inequality is a linear inequality in one variable. For instance, 2x ( 3 > 4 is a linear inequality in x. In the following examples, pay special attention to the steps in which the inequality symbol is reversed. Remember that when you multiply or divide by a negative number, you must reverse the inequality symbol. Example 2 Solving a Linear Inequality Solve 5x ! 7 > 3x ( 9. Solution Checking the solution set of an inequality is not as simple as checking the solutions of an equation. You can, however, get an indication of the validity of a solution set by substituting a few convenient values of x. For instance, in Example 2, try substituting x # 5 and x # 10 into the original inequality. 5x ! 7 > 3x ( 9 Write original inequality. 2x ! 7 > 9 Subtract 3x from each side. 2x > 16 Add 7 to each side. x > 8 Divide each side by 2. The solution set is all real numbers that are greater than 8, which is denoted by (8, ''. The graph of this solution set is shown in Figure A.14. Note that a parenthesis at 8 on the real number line indicates that 8 is not part of the solution set. x 6 7 8 9 10 Solution interval: (8, '' FIGURE A.14 Now try Exercise 35. Example 3 Solving a Linear Inequality Solve 1 ! 32 x & x ! 4. Graphical Solution Algebraic Solution 3x 1! & x!4 2 Write original inequality. 2 ! 3x & 2x ! 8 Multiply each side by 2. 2 ! 5x & !8 Subtract 2x from each side. !5x & !10 x % 2 Use a graphing utility to graph y1 # 1 ! 32 x and y2 # x ! 4 in the same viewing window. In Figure A.16, you can see that the graphs appear to intersect at the point (2, !2'. Use the intersect feature of the graphing utility to confirm this. The graph of y1 lies above the graph of y2 to the left of their point of intersection, which implies that y1 & y2 for all x % 2. Subtract 2 from each side. Divide each side by !5 and reverse the inequality. The solution set is all real numbers that are less than or equal to 2, which is denoted by (! ', 2*. The graph of this solution set is shown in Figure A.15. Note that a bracket at 2 on the real number line indicates that 2 is part of the solution set. 2 −5 7 y1 = 1 − 32 x x 0 1 2 Solution interval: (! ', 2* FIGURE A.15 Now try Exercise 37. 3 y2 = x − 4 −6 4 FIGURE A.16 A66 Appendix A Review of Fundamental Concepts of Algebra Sometimes it is possible to write two inequalities as a double inequality. For instance, you can write the two inequalities !4 % 5x ! 2 and 5x ! 2 < 7 more simply as !4 % 5x ! 2 < 7. Double inequality This form allows you to solve the two inequalities together, as demonstrated in Example 4. Example 4 Solving a Double Inequality To solve a double inequality, you can isolate x as the middle term. !3 % 6x ! 1 < 3 Original inequality !3 ( 1 % 6x ! 1 ( 1 < 3 ( 1 Add 1 to each part. !2 % 6x < 4 Simplify. !2 6x 4 < % 6 6 6 Divide each part by 6. ! 1 2 % x< 3 3 Simplify. The solution set is all real numbers that are greater than or equal to ! 13 and less than 2 1 2 3 , which is denoted by +! 3 , 3 '. The graph of this solution set is shown in Figure A.17. − 13 2 3 x −1 0 1 Solution interval: +! 13, 23 ' FIGURE A.17 Now try Exercise 47. The double inequality in Example 4 could have been solved in two parts, as follows. !3 % 6x ! 1 !2 % 6x 1 ! % x 3 and 6x ! 1 < 3 6x < 4 x < 2 3 The solution set consists of all real numbers that satisfy both inequalities. In other words, the solution set is the set of all values of x for which ! 2 1 %x< . 3 3 When combining two inequalities to form a double inequality, be sure that the inequalities satisfy the Transitive Property. For instance, it is incorrect to combine the inequalities 3 < x and x % !1 as 3 < x % !1. This “inequality” is wrong because 3 is not less than !1. Appendix A.6 A67 Linear Inequalities in One Variable Inequalities Involving Absolute Values Solving an Absolute Value Inequality T E C H N O LO G Y Let x be a variable or an algebraic expression and let a be a real number such that a & 0. A graphing utility can be used to identify the solution set of the graph of an inequality. For instance, to find the solution set of x ! 5 < 2 (see Example 5a), rewrite the inequality as x ! 5 ! 2 < 0, enter , , ,, !a < x < a. Double inequality ,, 2. The solutions of x > a are all values of x that are less than !a or greater than a. , ,x, > a Y1 " abs / X ! 50 ! 2 6 if and only if x < a , and press the graph key. The graph should look like the one shown below. ,, 1. The solutions of x < a are all values of x that lie between !a and a. if and only if x < !a or x > a. Compound inequality These rules are also valid if < is replaced by ≤ and > is replaced by ≥. Example 5 Solving an Absolute Value Inequality Solve each inequality. −1 10 , , , a. x ! 5 < 2 , b. x ( 3 & 7 Solution a. −4 ,x ! 5, < 2 Write original inequality. !2 < x ! 5 < 2 Notice that the graph is below the x-axis on the interval /3, 70. Write equivalent inequalities. !2 ( 5 < x ! 5 ( 5 < 2 ( 5 Add 5 to each part. 3 < x < 7 Simplify. The solution set is all real numbers that are greater than 3 and less than 7, which is denoted by (3, 7'. The graph of this solution set is shown in Figure A.18. b. ,x ( 3, & 7 Write original inequality. x ( 3 % !7 or x ( 3 ! 3 % !7 ! 3 x(3 & 7 x ( 3!3 & 7!3 , , Subtract 3 from each side. x & 4 x % !10 Note that the graph of the inequality x ! 5 < 2 can be described as all real numbers within two units of 5, as shown in Figure A.18. Write equivalent inequalities. Simplify. The solution set is all real numbers that are less than or equal to !10 or greater than or equal to 4. The interval notation for this solution set is (! ', !10* % +4, ''. The symbol % is called a union symbol and is used to denote the combining of two sets. The graph of this solution set is shown in Figure A.19. 2 units 2 units 7 units 7 units x x 2 3 4 5 6 7 8 ,x ! 5, < 2: Solutions lie inside (3, 7'. FIGURE A.18 Now try Exercise 61. −12 −10 −8 −6 −4 −2 0 2 4 6 ,x ( 3, & 7: Solutions lie outside (!10, 4'. FIGURE A.19 A68 Appendix A Review of Fundamental Concepts of Algebra Applications A problem-solving plan can be used to model and solve real-life problems that involve inequalities, as illustrated in Example 6. Example 6 Comparative Shopping You are choosing between two different cell phone plans. Plan A costs $49.99 per month for 500 minutes plus $0.40 for each additional minute. Plan B costs $45.99 per month for 500 minutes plus $0.45 for each additional minute. How many additional minutes must you use in one month for plan B to cost more than plan A? Solution Verbal Model: Labels: Monthly cost for plan B > Monthly cost for plan A Minutes used (over 500) in one month # m Monthly cost for plan A # 0.40m ( 49.99 Monthly cost for plan B # 0.45m ( 45.99 (minutes) (dollars) (dollars) Inequality: 0.45m ( 45.99 > 0.40m ( 49.99 0.05m > 4 m > 80 minutes Plan B costs more if you use more than 80 additional minutes in one month. Now try Exercise 111. Example 7 Accuracy of a Measurement You go to a candy store to buy chocolates that cost $9.89 per pound. The scale that is used in the store has a state seal of approval that indicates the scale is accurate to 1 within half an ounce (or 32 of a pound). According to the scale, your purchase weighs one-half pound and costs $4.95. How much might you have been undercharged or overcharged as a result of inaccuracy in the scale? Solution Let x represent the true weight of the candy. Because the scale is accurate 1 to within half an ounce (or 32 of a pound), the difference between the exact weight , , (x' and the scale weight ( 12 ' is less than or equal to 321 of a pound. That is, x ! 12 ≤ You can solve this inequality as follows. 1 1 ! 32 % x!2 % 15 32 % x % 1 32 . 1 32 17 32 0.46875 % x % 0.53125 In other words, your “one-half pound” of candy could have weighed as little as 0.46875 pound (which would have cost $4.64) or as much as 0.53125 pound (which would have cost $5.25). So, you could have been overcharged by as much as $0.31 or undercharged by as much as $0.30. Now try Exercise 125. Appendix A.6 A.6 EXERCISES Linear Inequalities in One Variable A69 See www.CalcChat.com for worked-out solutions to odd-numbered exercises. VOCABULARY: Fill in the blanks. 1. The set of all real numbers that are solutions of an inequality is the ________ ________ of the inequality. 2. The set of all points on the real number line that represents the solution set of an inequality is the ________ of the inequality. 3. To solve a linear inequality in one variable, you can use the properties of inequalities, which are identical to those used to solve equations, with the exception of multiplying or dividing each side by a ________ number. 4. Two inequalities that have the same solution set are ________. 5. It is sometimes possible to write two inequalities as one inequality, called a ________ inequality. 6. The symbol % is called a ________ symbol and is used to denote the combining of two sets. SKILLS AND APPLICATIONS In Exercises 7–14, (a) write an inequality that represents the interval and (b) state whether the interval is bounded or unbounded. 7. 9. 11. 13. +0, 9' +!1, 5* (11, '' (! ', !2' (!7, 4' (2, 10* +!5, '' (! ', 7* 8. 10. 12. 14. (a) x −4 −3 −2 −1 0 1 2 3 4 x 3 4 5 x −2 −1 0 0 1 1 2 3 4 5 x 2 3 4 5 (e) x −3 −2 −1 0 1 2 3 4 5 6 (f) x −5 −4 −3 −2 −1 0 1 2 3 4 5 (g) x −3 −2 −1 0 1 2 3 4 5 6 (h) x 4 15. 17. 19. 21. 5 x < 3 !3 < x % 4 x < 3 !1 % x % 52 ,, 6 7 16. 18. 20. 22. 8 x & 5 0 % x % 92 x > 4 !1 < x < 52 ,, , , , , 27. x ! 10 & 3 28. 2x ! 3 < 15 6 (d) −1 x!2 < 2 4 26. !5 < 2x ! 1 % 1 6 (c) −3 25. 0 < 5 (b) 2 Inequality 23. 5x ! 12 > 0 24. 2x ( 1 < !3 In Exercises 15–22, match the inequality with its graph. [The graphs are labeled (a)–(h).] −5 In Exercises 23–28, determine whether each value of x is a solution of the inequality. (a) (c) (a) (c) (a) (c) (a) (c) (a) (c) (a) (c) Values x#3 (b) 5 x#2 (d) x#0 (b) x # !4 (d) x#4 (b) x#0 (d) 1 x # !2 (b) 4 x#3 (d) x # 13 (b) x # 14 (d) x # !6 (b) x # 12 (d) x # !3 x # 32 x # ! 14 x # ! 32 x # 10 x # 72 x # ! 52 x#0 x # !1 x#9 x#0 x#7 In Exercises 29–56, solve the inequality and sketch the solution on the real number line. (Some inequalities have no solutions.) 29. 4x < 12 31. !2x > !3 33. 35. 37. 39. 41. 43. 45. 47. 48. 49. 50. 30. 10x < ! 40 32. !6x > 15 x!5 & 7 34. 2x ( 7 < 3 ( 4x 36. 2x ! 1 & 1 ! 5x 38. 4 ! 2x < 3(3 ! x' 40. 3 42. 4x ! 6 % x ! 7 1 5 ( 8x ( 1 ' 3x ( 44. & 2 2 3.6x ( 11 & !3.4 46. 1 < 2x ( 3 < 9 !8 % ! (3x ( 5' < 13 !8 % 1 ! 3(x ! 2' < 13 0 % 2 ! 3(x ( 1' < 20 x ( 7 % 12 3x ( 1 & 2 ( x 6x ! 4 % 2 ( 8x 4(x ( 1' < 2x ( 3 3 ( 27 x > x ! 2 9x ! 1 < 34(16x ! 2' 15.6 ! 1.3x < !5.2 A70 Appendix A Review of Fundamental Concepts of Algebra 2x ! 3 < 4 3 3 1 53. > x ( 1 > 4 4 51. !4 < 55. 3.2 % 0.4x ! 1 % 4.4 52. 0 % , x(3 < 5 2 x < 1 3 1.5x ( 6 56. 4.5 > > 10.5 2 54. !1 < 2 ! In Exercises 57–72, solve the inequality and sketch the solution on the real number line. (Some inequalities have no solution.) ,, 57. x < 5 x 59. > 1 2 61. x ! 5 < !1 63. x ! 20 % 6 65. 3 ! 4x & 9 x!3 67. &4 2 69. 9 ! 2x ! 2 < !1 71. 2 x ( 10 & 9 ,, , , , , , , ,, , , , , ,, 58. x & 8 x 60. > 3 5 ,, , , , 6x > 12 5 ! 2x & 1 4(x ! 3' % 8 ! x x ! 8 % 14 2 x ( 7 & 13 , , , , , , 83. 84. 85. 86. 87. 88. y # , , 1 2x , , (1 (a) (a) (a) (a) (a) x −3 −2 −1 0 1 2 3 2 3 x , , , Inequalities y & 1 (b) y % 5 (b) 0 % y % 3 (b) !1 % y % 3 (b) y % 2 (b) (a) y % 4 y y y y y % & & % & 90. "x ! 10 92. "3 ! x 4 6x ( 15 94. " −2 −1 0 1 x 5 6 7 8 9 10 11 12 13 14 0 1 2 3 x −7 0 0 0 0 4 (b) y & 1 In Exercises 89–94, find the interval(s) on the real number line for which the radicand is nonnegative. 89. "x ! 5 91. "x ( 3 4 7 ! 2x 93. " 97. 100. GRAPHICAL ANALYSIS In Exercises 83– 88, use a graphing utility to graph the equation. Use the graph to approximate the values of x that satisfy each inequality. Equation y # 2x ! 3 y # 23x ( 1 y # ! 12x ( 2 y # !3x ( 8 y# x!3 , In Exercises 97–104, use absolute value notation to define the interval (or pair of intervals) on the real number line. 4 3x ! 1 % 5 20 < 6x ! 1 3(x ( 1' < x ( 7 2x ( 9 > 13 1 2 x ( 1 % 3 , , 99. , ,, ,, 74. 76. 78. 80. 82. , , , −3 GRAPHICAL ANALYSIS In Exercises 73 – 82, use a graphing utility to graph the inequality and identify the solution set. 73. 75. 77. 79. 81. , 98. 62. x ! 7 < !5 64. x ! 8 & 0 66. 1 ! 2x < 5 2x 68. 1 ! < 1 3 70. x ( 14 ( 3 > 17 72. 3 4 ! 5x % 9 , , , 95. THINK ABOUT IT The graph of x ! 5 < 3 can be described as all real numbers within three units of 5. Give a similar description of x ! 10 < 8. 96. THINK ABOUT IT The graph of x ! 2 > 5 can be described as all real numbers more than five units from 2. Give a similar description of x ! 8 > 4. 101. 102. 103. 104. −6 −5 −4 −3 −2 −1 All real numbers within 10 units of 12 All real numbers at least five units from 8 All real numbers more than four units from !3 All real numbers no more than seven units from !6 In Exercises 105–108, use inequality notation to describe the subset of real numbers. 105. A company expects its earnings per share E for the next quarter to be no less than $4.10 and no more than $4.25. 106. The estimated daily oil production p at a refinery is greater than 2 million barrels but less than 2.4 million barrels. 107. According to a survey, the percent p of U.S. citizens that now conduct most of their banking transactions online is no more than 45%. 108. The net income I of a company is expected to be no less than $239 million. PHYSIOLOGY In Exercises 109 and 110, use the following information. The maximum heart rate of a person in normal health is related to the person’s age by the equation r " 220 ! A, where r is the maximum heart rate in beats per minute and A is the person’s age in years. Some physiologists recommend that during physical activity a sedentary person should strive to increase his or her heart rate to at least 50% of the maximum heart rate, and a highly fit person should strive to increase his or her heart rate to at most 85% of the maximum heart rate. (Source: American Heart Association) 109. Express as an interval the range of the target heart rate for a 20-year-old. 110. Express as an interval the range of the target heart rate for a 40-year-old. Appendix A.6 111. JOB OFFERS You are considering two job offers. The first job pays $13.50 per hour. The second job pays $9.00 per hour plus $0.75 per unit produced per hour. Write an inequality yielding the number of units x that must be produced per hour to make the second job pay the greater hourly wage. Solve the inequality. 112. JOB OFFERS You are considering two job offers. The first job pays $3000 per month. The second job pays $1000 per month plus a commission of 4% of your gross sales. Write an inequality yielding the gross sales x per month for which the second job will pay the greater monthly wage. Solve the inequality. 113. INVESTMENT In order for an investment of $1000 to grow to more than $1062.50 in 2 years, what must the annual interest rate be? +A # P(1 ( rt'* 114. INVESTMENT In order for an investment of $750 to grow to more than $825 in 2 years, what must the annual interest rate be? +A # P(1 ( rt'* 115. COST, REVENUE, AND PROFIT The revenue from selling x units of a product is R # 115.95x. The cost of producing x units is C # 95x ( 750. To obtain a profit, the revenue must be greater than the cost. For what values of x will this product return a profit? 116. COST, REVENUE, AND PROFIT The revenue from selling x units of a product is R # 24.55x. The cost of producing x units is C # 15.4x ( 150,000. To obtain a profit, the revenue must be greater than the cost. For what values of x will this product return a profit? 117. DAILY SALES A doughnut shop sells a dozen doughnuts for $4.50. Beyond the fixed costs (rent, utilities, and insurance) of $220 per day, it costs $2.75 for enough materials (flour, sugar, and so on) and labor to produce a dozen doughnuts. The daily profit from doughnut sales varies from $60 to $270. Between what levels (in dozens) do the daily sales vary? 118. WEIGHT LOSS PROGRAM A person enrolls in a diet and exercise program that guarantees a loss of at least 112 pounds per week. The person’s weight at the beginning of the program is 164 pounds. Find the maximum number of weeks before the person attains a goal weight of 128 pounds. 119. DATA ANALYSIS: IQ SCORES AND GPA The admissions office of a college wants to determine whether there is a relationship between IQ scores x and grade-point averages y after the first year of school. An equation that models the data the admissions office obtained is y # 0.067x ! 5.638. (a) Use a graphing utility to graph the model. (b) Use the graph to estimate the values of x that predict a grade-point average of at least 3.0. Linear Inequalities in One Variable A71 120. DATA ANALYSIS: WEIGHTLIFTING You want to determine whether there is a relationship between an athlete’s weight x (in pounds) and the athlete’s maximum bench-press weight y (in pounds). The table shows a sample of data from 12 athletes. Athlete’s weight, x Bench-press weight, y 165 184 150 210 196 240 202 170 185 190 230 160 170 185 200 255 205 295 190 175 195 185 250 155 (a) Use a graphing utility to plot the data. (b) A model for the data is y # 1.3x ! 36. Use a graphing utility to graph the model in the same viewing window used in part (a). (c) Use the graph to estimate the values of x that predict a maximum bench-press weight of at least 200 pounds. (d) Verify your estimate from part (c) algebraically. (e) Use the graph to write a statement about the accuracy of the model. If you think the graph indicates that an athlete’s weight is not a particularly good indicator of the athlete’s maximum bench-press weight, list other factors that might influence an individual’s maximum bench-press weight. 121. TEACHERS’ SALARIES The average salaries S (in thousands of dollars) for elementary school teachers in the United States from 1990 through 2005 are approximated by the model S # 1.09t ( 30.9, 0 % t % 15 where t represents the year, with t # 0 corresponding to 1990. (Source: National Education Association) (a) According to this model, when was the average salary at least $32,500, but not more than $42,000? (b) According to this model, when will the average salary exceed $54,000? Appendix A Review of Fundamental Concepts of Algebra 122. EGG PRODUCTION The numbers of eggs E (in billions) produced in the United States from 1990 through 2006 can be modeled by E # 1.52t ( 68.0, 0 % t % 16 123. 124. 125. 126. 127. where t represents the year, with t # 0 corresponding to 1990. (Source: U.S. Department of Agriculture) (a) According to this model, when was the annual egg production 70 billion, but no more than 80 billion? (b) According to this model, when will the annual egg production exceed 100 billion? GEOMETRY The side of a square is measured as 1 10.4 inches with a possible error of 16 inch. Using these measurements, determine the interval containing the possible areas of the square. GEOMETRY The side of a square is measured as 24.2 centimeters with a possible error of 0.25 centimeter. Using these measurements, determine the interval containing the possible areas of the square. ACCURACY OF MEASUREMENT You stop at a self-service gas station to buy 15 gallons of 87-octane gasoline at $2.09 a gallon. The gas pump is accurate to 1 within 10 of a gallon. How much might you be undercharged or overcharged? ACCURACY OF MEASUREMENT You buy six T-bone steaks that cost $14.99 per pound. The weight that is listed on the package is 5.72 pounds. The scale that weighed the package is accurate to within 12 ounce. How much might you be undercharged or overcharged? TIME STUDY A time study was conducted to determine the length of time required to perform a particular task in a manufacturing process. The times required by approximately two-thirds of the workers in the study satisfied the inequality , , t ! 15.6 < 1 1.9 where t is time in minutes. Determine the interval on the real number line in which these times lie. 128. HEIGHT The heights h of two-thirds of the members of a population satisfy the inequality , , 130. MUSIC Michael Kasha of Florida State University used physics and mathematics to design a new classical guitar. The model he used for the frequency of the vibrations on a circular plate was v # (2.6t)d 2'"E)-, where v is the frequency (in vibrations per second), t is the plate thickness (in millimeters), d is the diameter of the plate, E is the elasticity of the plate material, and - is the density of the plate material. For fixed values of d, E, and -, the graph of the equation is a line (see figure). Frequency (vibrations per second) A72 v 700 600 500 400 300 200 100 1 2 (a) Estimate the frequency when the plate thickness is 2 millimeters. (b) Estimate the plate thickness when the frequency is 600 vibrations per second. (c) Approximate the interval for the plate thickness when the frequency is between 200 and 400 vibrations per second. (d) Approximate the interval for the frequency when the plate thickness is less than 3 millimeters. EXPLORATION TRUE OR FALSE? In Exercises 131 and 132, determine whether the statement is true or false. Justify your answer. 131. If a, b, and c are real numbers, and a % b, then ac % bc. 132. If !10 % x % 8, then !10 & !x and !x & !8. , , , , 133. Identify the graph of the inequality x ! a & 2. (a) (b) x a−2 2 x a−2 a+2 a x 2−a where h is measured in inches. Determine the interval on the real number line in which these heights lie. 129. METEOROLOGY An electronic device is to be operated in an environment with relative humidity h in the interval defined by h ! 50 % 30. What are the minimum and maximum relative humidities for the operation of this device? t 4 Plate thickness (in millimeters) (c) h ! 68.5 %1 2.7 3 a a+2 (d) x 2−a 2+a 2 2+a 134. Find sets of values of a, b, and c such that 0 % x % 10 is a solution of the inequality ax ! b % c. 135. Give an example of an inequality with an unbounded solution set. , , 136. CAPSTONE Describe any differences between properties of equalities and properties of inequalities. PreCalculus Practice Test 2014-2015 (Due August 14, 2014) 1. The length of a rectangle is ( ) centimeters and the width is ( the area, in square centimeters, of the rectangle. ) centimeters. Which expression represents 2. Find the area of the given triangle. 23 4 10 9 3. The area of a triangle is 78 square feet. Find the length of the base if the height is 13 feet. 4. Find the area, in mm, of the shaded region comprised of a rectangle and a right triangle. 8 4 6 10 5. Given: 1 q p 2 m What is n ? 6. In the figure below, line m is parallel to line n. 1 m 2 3 n 5 4 Determine whether the folloing statements are true or false. True or False True or False True or False True or False True or False 7. In the accompanying diagram, parallel lines l and m are cut by a transversal t. t 1 l 2 m Determine whether the following statements are true or false. is the complement of True or False . and are right angles. is the supplement of . True or False True or False True or False True or False 8. If the area of a circle is 49 9. In the triangles below, inches , find the circumference of the circle. . R (3x)° Q S U (5x–8)° (8x+12)° T V What is the value of x? 10. A right triangle’s hypotenuse has length 6. If one leg has length 5, what is the length of the other leg? 11. Find the value of x. Write your answer in simplest radical form. 10 45º x 12. If in the right triangle below, what is the values of b? a b c 30 º 13. What is the area, in square inches (in), of the triangle below? 16 in. 16 in. 16 in. 14. Simplify: (x + 2)(5x – 2) – (–3x – 8) 15. Simplify: 16. Evaluate. 17. Given the polynomial, 7x + 5x3 – 35x2 – 49, identify the leading coefficient, degree and constant. 18. Evaluate. 19. Simplify. 20. Simplify. 21. Rationalize and simplify. 22. Write the rational expression in simplest form. 23. Factor the sum of two cubes. 24. Factor completely. 25. Factor by grouping. 20x3 – 25x2 – 16x + 20 26. Multiply. 27. Simplify. 28. Solve. 29. Solve. 30. Solve. 31. Write the interval [–9, 1) in inquality notation, sketch the subset on a real number line, and state whether the interval is bound or unbound. 32. What is the complete solution to the equation 33. Perform the opertation. 34. Write the rational expression in simplest form. 35. Solve the equation. 36. Solve. 37. Solve. 38. Solve. 39. Simplify to lowest terms. 40. Which is the factored form of ? PreCalculus Practice Test 2014-2015 (Due August 14, 2014) Answer Section 1. 2. 3. 4. 5. 6. 7. 8. 9. 18 units2 12 feet 68 113º T,F,T,T,T F,F,F,T,T C = 14 inches 11 10. 11 11. 10 12. 11 13. 64 14. 15. 16. 3 17. 5, 3, –49 18. 64 19. 20. 21. 22. 23. 24. 25. (5x 2 – 4)(4x – 5) 26. 27. 28. 13x = 17 29. x = 0, + 3 2 , 30. x = 128 31. –9 –8 –7 –6 32. x = –2; x = –4 –5 –4 –3 –2 –1 0 1 33. 34. 35. , –6 36. x = 21 37. x > 20 or x < –14 38. x = –5, 2 39. 40. 2(a – 3b)(a – 3b)