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Chapter 1 A1 D D A A D A D D 5. The phrase twice the sum of a number squared and 6 could be written as 2n2 6. 6. The reflexive property of equality states that if a b then b a. 7. The absolute value of a number is its distance from 0 on the number line. 8. If the absolute value of any expression is equal to a negative number, then the solution is the empty set. 9. When adding or subtracting a negative number to both sides of an inequality, the inequality symbol must be reversed. 10. Writing a solution in the form {x| x 5} is called set builder notation. 11. If a compound inequality contains the word “or”, the solution will be the intersection of the solution sets of the two inequalities. 12. If |3x 1| 10, then 3x 1 10 and 3a 1 10. After you complete Chapter 1 A 4. The commutative property is true for addition and multiplication only. A Chapter 1 3 Glencoe Algebra 2 • For those statements that you mark with a D, use a piece of paper to write an example of why you disagree. • Did any of your opinions about the statements change from the first column? time drops minutes drops . . per milliliter. milliliters per minute. Chapter 1 5 Answers Glencoe Algebra 2 Sample answer: Please excuse my dear Aunt Sally. (parentheses; exponents; multiplication and division; addition and subtraction) 4. Think of a phrase or sentence to help you remember the order of operations. Remember What You learned operations, different people might get different answers. 3. Why is it important for everyone to use the same order of operations for evaluating expressions? Sample answer: If everyone did not use the same order of Multiply the difference of 13 and 5 by the sum of 9 and 21. Add the result to 10. Then divide what you get by 2. [(13 5)(9 21) 10] 2 2. Read the following instructions. Then use grouping symbols to show how the instructions can be put in the form of a mathematical expression. c. {14 [8 (3 12)2]} (63 100) (3 12) b. 9 [5(8 6) 2(10 7)] (8 6) and (10 7) a. [(3 22) 8] 4 (3 22) 1. There is a customary order for grouping symbols. Brackets are used outside of parentheses. Braces are used outside of brackets. Identify the innermost expression(s) in each of the following expressions. Read the Lesson 8 60 and is measured in of solution and is measured in and is measured in and is measured in drop factor volume flow rate • Write the expression that a nurse would use to calculate the flow rate of an IV if a doctor orders 1350 milliliters of IV saline to be given over 8 hours, with a drop factor of 20 drops per milliliter. Do not find the value of this expression. 1350 20 t represents d represents the F represents the A D The order of operations must be followed so that every expression will have only one value. Chapter Resources V represents the 3. All real numbers are in the set of rational numbers. 2. 1. Algebraic expressions contain at least one variable. Statement Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. STEP 2 A or D Answers • Reread each statement and complete the last column by entering an A or a D. STEP 2 STEP 1 A, D, or NS the quantity that each of the variables in this formula represents and the units in which each is measured. • Nurses use the formula F to control the flow rate for IVs. Name t Vd Read the introduction to Lesson 1-1 in your textbook. ____________ PERIOD _____ 8:03 AM • Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure). • Decide whether you Agree (A) or Disagree (D) with the statement. Expressions and Formulas Lesson Reading Guide Get Ready for the Lesson 1-1 NAME ______________________________________________ DATE 5/19/06 • Read each statement. Before you begin Chapter 1 Equations and Inequalities Anticipation Guide ____________ PERIOD _____ A2-01-873971 STEP 1 1 NAME ______________________________________________ DATE Lesson 1-1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A1 (Lesson 1-1) Glencoe Algebra 2 Chapter 1 Expressions and Formulas Study Guide and Intervention 5. (5 23)2 52 144 8. (7 32)2 62 40 4. 9(32 6) 135 A2 7. 6 2 6 26. b c 4 d 15 25. a b c 7.45 Chapter 1 23. cd 4 22. 3 b 21 b d 20. (b c)2 4a 81.8 19. ac bd 31.3 dc 17. 5(6c 8b 10d) 215 16. 49.2 ab d Glencoe Algebra 2 27. d 8.7 a bc 24. d(a c) 6.1 21. 6b 5c 54.4 a d 18. 6 c2 1 bd Evaluate each expression if a 8.2, b 3, c 4, and d . 1 2 13. 8(42 8 32) 240 4 6 9 3 15 15. 82 642 14. 461 24 12. 6(7) 4 4 5 38 11. 14 (8 20 2) 7 10. 12 6 3 2(4) 6 9. 20 22 6 11 6. 52 18 2 34.25 1 4 3. 2 (4 2)3 6 4 Example np 500 To calculate the number of reams of paper needed to print n (172)(25) 500 43,000 500 Chapter 1 7 Glencoe Algebra 2 4. A person’s basal metabolic rate (or BMR) is the number of calories needed to support his or her bodily functions for one day. The BMR of an 80-year-old man is given by the formula BMR 12w (0.02)(6)12w, where w is the man’s weight in pounds. What is the BMR of an 80-year-old man who weighs 170 pounds? 1795 calories 3. Sarah takes 40 breaths to blow up the beach ball. What is the average volume of air per breath? about 1250 cm3 sphere and r is its radius. What is the volume of the beach ball in cubic centimeters? (Use 3.14 for .) 50,015 cm3 2. The volume of a sphere is given by the formula V r3, where V is the volume of the 4 3 1. Her beach ball has a radius of 9 inches. First she converts the radius to centimeters using the formula C 2.54I, where C is a length in centimeters and I is the same length in inches. How many centimeters are there in 9 inches? 22.86 cm For a science experiment, Sarah counts the number of breaths needed for her to blow up a beach ball. She will then find the volume of the beach ball in cubic centimeters and divide by the number of breaths to find the average volume of air per breath. For Exercises 1–3, use the following information. Exercises You cannot buy 8.6 reams of paper. You will need to buy 9 reams to print 172 copies. 8.6 r np 500 Substitute n 172 and p 25 into the formula r . is the number of reams needed. How many reams of paper must you buy to print 172 copies of a 25-page booklet? copies of a booklet that is p pages long, you can use the formula r , where r Answers 16 23 4 12 2. 11 (3 2)2 14 1. 14 (6 2) 17 Find the value of each expression. Replace each variable with the given value. 3x2 x(y 5) 3 (3)2 3(0.5 5) 3 (9) 3(4.5) 27 13.5 13.5 Evaluate 3x2 x(y 5) if x 3 and y 0.5. (continued) Formulas A formula is a mathematical sentence that uses variables to express the relationship between certain quantities. If you know the value of every variable except one in a formula, you can use substitution and the order of operations to find the value of the unknown variable. Expressions and Formulas Study Guide and Intervention ____________ PERIOD _____ 8:03 AM Exercises [18 (6 4)] 2 [18 10] 2 82 4 Evaluate [18 (6 4)] 2. Example 2 Simplify the expressions inside grouping symbols. Evaluate all powers. Do all multiplications and divisions from left to right. Do all additions and subtractions from left to right. 1-1 NAME ______________________________________________ DATE 5/19/06 Example 1 Order of Operations 1. 2. 3. 4. ____________ PERIOD _____ A2-01-873971 Order of Operations 1-1 NAME ______________________________________________ DATE Lesson 1-1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A2 (Lesson 1-1) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Expressions and Formulas Skills Practice 6(7 5) 6. 4 1 5. [9 10(3)] 3 1 2 8. [3(5) 128 22]5 85 A3 25 2 rv3 s 20. 7s 2v 22 19. 9r2 (s2 1)t 105 Chapter 1 8 Glencoe Algebra 2 for a given temperature in degrees Fahrenheit. What is the temperature in degrees Celsius when the temperature is 68 degrees Fahrenheit? 20C 5 9 22. TEMPERATURE The formula C (F 32) gives the temperature in degrees Celsius 21. TEMPERATURE The formula K C 273 gives the temperature in kelvins (K) for a given temperature in degrees Celsius. What is the temperature in kelvins when the temperature is 55 degrees Celsius? 328 K 2w r 18. 0 2 17. w[t (t r)] 16. 4 3v t 5s t 14. s2r wt 3 13. (4s)2 144 8(13 37) 3 4 (8)2 9 5 c e 22. 2ab2 (d 3 c) 67 ac4 d 20. 2 22 18. ac3 b2de 70 d(b c) 16. 12 ac 14. (c d)b 8 Chapter 1 9 Answers Glencoe Algebra 2 25. AGRICULTURE Faith owns an organic apple orchard. From her experience the last few seasons, she has developed the formula P 20x 0.01x2 240 to predict her profit P in dollars this season if her trees produce x bushels of apples. What is Faith’s predicted profit this season if her orchard produces 300 bushels of apples? $4860 24. PHYSICS The formula h 120t 16t2 gives the height h in feet of an object t seconds after it is shot upward from Earth’s surface with an initial velocity of 120 feet per second. What will the height of the object be after 6 seconds? 144 ft Fahrenheit for a given temperature in degrees Celsius. What is the temperature in degrees Fahrenheit when the temperature is 40 degrees Celsius? 40F 23. TEMPERATURE The formula F C 32 gives the temperature in degrees 21. 9bc 141 1 e 19. b[a (c d) 2 ] 206 17. (b de)e2 1 ab c 15. d2 12 13. ab2 d 45 1 3 12. (1)2 4(9) 53 59 Evaluate each expression if a , b 8, c 2, d 3, and e . 11. 32 6 1 4 10. [5 5(3)] 5 8. [4(5 3) 2(4 8)] 16 1 7. 18 {5 [34 (17 11)]} 41 1 2 6. (2)3 (3)(8) (5)(10) 18 5. 20 (5 3) 52(3) 85 9. [6 42] 5 4. 12 [20 2(62 3 22)] 88 3. 1 2 3(4) 2 3 Answers 15. 2(3r w) 7 12. s 2r 16v 1 10. 2st 4rs 84 11. w(s r) 2 9. 6r 2s 0 Evaluate each expression if r 1, s 3, t 12, v 0, and w . 7. (168 7)32 43 152 3 4. 5 3(2 12 2) 7 3. (3 8)2(4) 3 97 2. 4(12 42) 16 1. 3(4 7) 11 20 ____________ PERIOD _____ 8:03 AM 7 2. 9 6 2 1 13 Expressions and Formulas Practice Find the value of each expression. 1-1 NAME ______________________________________________ DATE 5/19/06 1. 18 2 3 27 ____________ PERIOD _____ A2-01-873971 Find the value of each expression. 1-1 NAME ______________________________________________ DATE Lesson 1-1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A3 (Lesson 1-1) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 R A4 Chapter 1 subtraction 10 + 3(4 + 6) = 4 10 17 min Glencoe Algebra 2 6. Use your formula to compute the number of minutes it would take to broil a 2 inch thick steak. Sample Answer: T 5(w 1) 12 or T 5w 7 5. Write a formula for T in terms of w. 5.7 5.0 4.4 4.0 3.6 3.3 3.0 2.8 2.6 Time (hours) 2 Chapter 1 11 Glencoe Algebra 2 You can make it to your aunt’s house on time, but won’t have enough gas to get home. the speeds needed to get to your aunt’s. Will you make it? 30 1 5 formula M S2 S, where S is your speed. Determine your fuel rate for 4. Cost depends on the cost of gasoline, the number of total miles of the trip, and your car’s fuel efficiency (mi/gal). The miles per gallon can be found using the About 8.5 gallons 3. How many gallons of gas can you buy? Now determine if you can afford enough gasoline to make the trip and return. You will miss the party for all speeds 50 mph and less, assuming an 11 A.M. departure. 2. For which speed(s), will you miss the surprise birthday party? 75 70 65 60 55 50 45 40 35 Speed (mph) and S is the speed. Determine travel time to your aunt’s house at various speeds. S D miles per hour (mph), T , where T is time in hours, D is the distance (200 miles), 1. A simple formula relates the travel time, depending on your average speed in Answers 3. GUESS AND CHECK Amanda received a worksheet from her teacher. Unfortunately, one of the operations in an equation was covered by a blot. What operation is hidden by the blot? 236 in2 r A steak has thickness w inches. Let T be the time it takes to broil the steak. It takes 12 minutes to broil a one inch thick steak. For every additional inch of thickness, the steak should be broiled for 5 more minutes. the following information. COOKING For Exercises 5 and 6, use 56 mi First determine at which speed you must travel to arrive by 3:00. 8:03 AM 2. GEOMETRY The formula for the area of a ring-shaped object is given by A (R2 r2), where R is the radius of the outer circle and r is the radius of the inner circle. If R 10 inches and r 5 inches, what is the area rounded to the nearest square inch? 10 6 10 12 or 10(6 12) Traveling on a Budget Enrichment ____________ PERIOD _____ You are traveling to your aunt’s house 200 miles away for a surprise birthday party. The party starts at 3 P.M. but you cannot leave from your house before 11 A.M. You must fill your gas tank before the trip. Gasoline is $3.50 per gallon and you have $30. Will you make it to the party and make it back home? 1-1 NAME ______________________________________________ DATE 5/19/06 number of miles that Rick can drive on g gallons of gasoline is given by m 21g. How many miles can Rick drive on $8 worth of gasoline? 3 d is given by g . The formula for the 4. GAS MILEAGE Rick has d dollars. The formula for the number of gallons of gasoline that Rick can buy with d dollars Expressions and Formulas Word Problem Practice ____________ PERIOD _____ A2-01-873971 1. ARRANGEMENTS The chairs in an auditorium are arranged into two rectangles. Both rectangles are 10 rows deep. One rectangle has 6 chairs per row and the other has 12 chairs per row. Write an expression for the total number of chairs in the auditorium. 1-1 NAME ______________________________________________ DATE Lesson 1-1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A4 (Lesson 1-1) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Properties of Real Numbers Lesson Reading Guide A5 Chapter 1 12 Glencoe Algebra 2 A commuter is someone who travels back and forth to work or another place, and the commutative property says you can switch the order when two numbers that are being added or multiplied. An association is a group of people who are connected or united, and the associative property says that you can switch the grouping when three numbers are added or multiplied. 4. How can the meanings of the words commuter and association help you to remember the difference between the commutative and associative properties? Sample answer: Remember What You Learned The quantities a, b, and c are used in the same order, but they are grouped differently on the two sides of the equation. The second equation uses the quantities in different orders on the two sides of the equation. So the second equation uses the Commutative Property of Multiplication. 3. Consider the equations (a b) c a (b c) and (a b) c c (a b). One of the equations uses the Associative Property of Multiplication and one uses the Commutative Property of Multiplication. How can you tell which property is being used in each equation? The first equation uses the Associative Property of Multiplication. natural numbers whole numbers integers N W Z {all nonterminating, nonrepeating decimals} {…, 3, 2, 1, 0, 1, 2, 3, …} {0, 1, 2, 3, 4, 5, 6, 7, 8, …} {1, 2, 3, 4, 5, 6, 7, 8, 9, …} Exercises 6 7 15 3 13 Glencoe Algebra 2 20. 0.02 Q, R 17. 2 I, R 5 14. 42 I, R 11. 4.1 7 Q, R 8. 26.1 Q, R Answers 19. 894,000 N, W, Z, Q, R 18. 33.3 Q, R Chapter 1 16. Q, R 8 13 13. 1 Z, Q, R 10. N, W, Z, Q, R 36 7. Q, R 9 3. 0 W, Z, Q, R 15. 11.2 Q, R 1 2 6. 34 Q, R 2. 81 Z, Q, R 12. N, W, Z, Q, R 5 25 9. I, R 5. 73 N, W, Z, Q, R 1. Q, R 4. 192.0005 Q, R naturals (N), wholes (W), integers (Z), rationals (Q), reals (R) rationals (Q), reals (R) 25 25 5 m Name the sets of numbers to which each number belongs. b. 11 a. 3 {all rationals and irrationals} {all numbers that can be represented in the form , where m and n are integers and n n is not equal to 0} Name the sets of numbers to which each number belongs. irrational numbers I Example rational numbers Q Answers Sample answer: (12 18) 45 30 45 75; 12 (18 45) 12 63 75 2. Write the Associative Property of Addition in symbols. Then illustrate this property by finding the sum 12 18 45 in two different ways. (a b) c a (b c); decimal because there is a block of digits, 57, that repeats forever, so this number is rational. The number 0.010010001… is a non-repeating decimal because, although the digits follow a pattern, there is no block of digits that repeats. So this number is an irrational number. 1. Refer to the Key Concepts box on page 11. The numbers 2.5 7 and 0.010010001… both involve decimals that “go on forever.” Explain why one of these numbers is rational and the other is irrational. Sample answer: 2.5 7 2.5757… is a repeating Read the Lesson coupon amounts or add the amounts for the scanned coupons and multiply the sum by 2. real numbers R 8:03 AM • Describe two ways of calculating the amount of money you saved by using coupons if your register slip is the one shown on page 11. Sample answer: Add all the individual from the total amount of purchases so that you save money by using coupons. • Why are all of the amounts listed on the register slip at the top of the page followed by negative signs? Sample answer: The amount of each coupon is subtracted ____________ PERIOD _____ All real numbers can be classified as either rational or irrational. The set of rational numbers includes several subsets: natural numbers, whole numbers, and integers. Properties of Real Numbers Study Guide and Intervention Real Numbers 1-2 NAME ______________________________________________ DATE 5/19/06 Read the introduction to Lesson 1-2 in your textbook. ____________ PERIOD _____ A2-01-873971 Get Ready for the Lesson 1-2 NAME ______________________________________________ DATE Lesson 1-2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A5 (Lesson 1-2) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Properties of Real Numbers Study Guide and Intervention A6 3 4 4a 3b a b 5. 12 3 4 10.2r 39.2s 6. 8(2.4r 3.1s) 6(1.5r 2.4s) 2 k j 5 4.2k 5.4j 3 5 1 2 190a 70b Chapter 1 23. 3(m z) 5(2m z) 13m 8z 21. (3g 3h) 5g 10h 2g 13h 19. 3x 5 2x 3 5x 2 Simplify each expression. 14 Glencoe Algebra 2 5 4 17. , 4 4 5 5 1 15 15. 15 15, Chapter 1 2j 7 Comm. () 3 4 4 15 1 2 15 Glencoe Algebra 2 26. (15d 3) (8 10d) 10d 3 1 3 24. 2x 3y (5x 3y 2z) 3x 2z 22. a2 a 4a 3a2 1 2a2 3a 1 20. x y z y x z 0 3 4 18. 3 3 , 16. 1.25 1.25, 0.8 Name the additive inverse and multiplicative inverse for each number. Assoc. () 14. (10b 12b) 7b (12b 10b) 7b Mult. Iden. 13. 0.6[25(0.5)] [0.6(25)]0.5 12. 15x(1) 15x Mult. Inv. Assoc. () 10. 2r (3r 4r) (2r 3r) 4r Add. Iden. 8. 3a 0 3a 11. 5y 1 5y1 Distributive 9. 2(r w) 2r 2w 25. 6(2 v) 4(2v 1) 8 2v 20 2n 2 16. (18 6n 12 3n) 3 40 105c 14. 2(15 45c) (12 18c) 5 6 1 4 p r 4 5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 18. 50(3a b) 20(b 2a) 8.75m 1 5 12. p r r p 3 4 12.7x 16 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 17. 14( j 2) 3j(4 7) 0.7x 9 15. (7 2.1x)3 2(3.5x 6) 140g 120h 13. 4(10g 80h) 20(10h 5g) 62.4e 39f 10. 9(7e 4f) 0.6(e 5f ) 11. 2.5m(12 8.5) 77 4p 7. 4(20 4p) (4 16p) 8. 5.5j 8.9k 4.7k 10.9j 9. 1.2(7x 5) (10 4.3x) 80g 26h 4. 10(6g 3h) 4(5g h) 51s 13t 2. 40s 18t 5t 11s 1 3. (4j 2k 6j 3k) 5 Commutative Property () Distributive Property Simplify. Comm. () 7. 3 x x 3 Answers 20a 1. 8(3a b) 4(2b a) Simplify each expression. Exercises 9x 3y 12y 0.9x 9x ( 0.9x) 3y 12y (9 ( 0.9))x (3 12)y 8.1x 15y 6. 30 I, R 5. 9 Z, Q, R Name the property illustrated by each equation. 4. N, W, Z, Q, R 12 3 2. 525 Z, Q, R 3. 0.875 Q, R 1. 34 N, W, Z, Q, R ____________ PERIOD _____ 8:03 AM Simplify 9x 3y 12y 0.9x. a(b c) ab ac and (b c)a ba ca Distributive Properties of Real Numbers Skills Practice Name the sets of numbers to which each number belongs. 1-2 NAME ______________________________________________ DATE 5/19/06 Example a (a) 0 (a) a Inverse 1 a If a is not zero, then a 1 a. a0a0a Identity 1 a (a b) c a (b c) a 1a1 a (a b) c a (b c) Associative Multiplication a bb a Addition abba Commutative Property For any real numbers a, b, and c Real Number Properties (continued) ____________ PERIOD _____ A2-01-873971 Properties of Real Numbers 1-2 NAME ______________________________________________ DATE Lesson 1-2 A1-A23 Page A6 (Lesson 1-2) Glencoe Algebra 2 Chapter 1 Properties of Real Numbers Practice 25 36 Q, R 6. 16 Z, Q, R I, R 2. 7 7. 35 Z, Q, R I, R 3. 2 Distributive 17. 5(x y) 5x 5y Comm. () 14. 3x 2y 3 2 x y Distributive Add. Iden. 18. 4n 0 4n Add. Inv. 15. (6 6)y 0y 12. 7n 2n (7 2)n Assoc. () 10. 7x (9x 8) (7x 9x) 8 A7 16 11 5 3 6 5 29. 2(4 2x y) 4(5 x y) 1 2 1 4 Chapter 1 16 1 1 false; counterexample: 5 4 5 4 a1 1b Glencoe Algebra 2 Chapter 1 Rationals Irrationals Real Numbers 3. VENN DIAGRAMS Make a Venn diagram that shows the relationship between natural numbers, integers, rational numbers, irrational numbers, and real numbers. Integers statement is true or false: If a b, it follows that a b . Explain your reasoning. = 7 10 Commutative Property of Multiplication 32. NUMBER THEORY Use the properties of real numbers to tell whether the following Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Natural Numbers 13y 30. x 12y (2x 12y) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 31. TRAVEL Olivia drives her car at 60 miles per hour for t hours. Ian drives his car at 50 miles per hour for (t 2) hours. Write a simplified expression for the sum of the distances traveled by the two cars. (110t 100) mi 12 8x 6y 1 5 28. (10a 15) (8 4a) 4a 1 27. 3(r 10s) 4(7s 2r) 5r 58s 26. 4c 2c (4c 2c) 4c 25. 8x 7y (3 6y) 8x y 3 5 6 5 , 6 35 24. 11a 13b 7a 3b 4a 16b 5 22. 5 6 20. 1.6 1.6, 0.625 23. 5x 3y 2x 3y 3x Simplify each expression. 11 11 21. , 16 16 10 7 2. MODELS What property of real numbers is illustrated by the figure below? Distributive Property 17 b c Answers Glencoe Algebra 2 c 25; it is a natural number. 7. a 7, b 24 c 245 or 75 ; it is not a natural number. 6. a 7, b 14 c 13; it is a natural number. 5. a 5, b 12 For each set of values for a and b, determine the value of c. State whether c is a natural number. a The lengths of the sides of the right triangle shown are related by the formula c2 = a2 b2. RIGHT TRIANGLES For Exercises 5–7, use the following information. I. always II. sometimes, · 2 2 2 Answers 19. 0.4 0.4, 2.5 Name the additive inverse and multiplicative inverse for each number. Mult. Inv. 16. 4y 1y 1 4 Assoc. () 13. 3(2x)y (3 2)(xy) Mult. Iden. 11. 5(3x y) 5(3x 1y) Comm. () 9. 5x (4y 3x) 5x (3x 4y) 8. 31.8 Q, R W, Z, Q, R 4. 0 8:03 AM Name the property illustrated by each equation. 5. N, W, Z, Q, R ____________ PERIOD _____ 4. NUMBER THEORY Consider the following two statements. I. The product of any two rational numbers is always another rational number. II. The product of two irrational numbers is always irrational. Determine if these statements are always, sometimes, or never true. Explain. Properties of Real Numbers Word Problem Practice 1. MENTAL MATH When teaching elementary students to multiply and learn place value, books often show that 54 8 (50 4) 8 (50 8) (4 8). What property is used? 1-2 NAME ______________________________________________ DATE 5/19/06 1. 6425 ____________ PERIOD _____ A2-01-873971 Name the sets of numbers to which each number belongs. 1-2 NAME ______________________________________________ DATE Lesson 1-2 A1-A23 Page A7 (Lesson 1-2) Glencoe Algebra 2 Enrichment Chapter 1 Does each number, a, have an inverse, a , such that a a a a i? The integer inverse of 3 is 3 since 3 3 0, and 0 is the identity for addition. But the set does not contain 3. Therefore, there is no inverse for 3. Inverse: A8 1 is the inverse of 1 since (1)(1) 1, and 1 is the identity. 1 is the inverse of 1 since (1)(1) 1, and 1 is the identity. Each member has an inverse. Inverse: Chapter 1 Glencoe Algebra 2 8. {rational numbers}, addition yes 7. {irrational numbers}, addition no 18 6. {1 , 2 , 3 , …}, multiplication no 5. {x, x2, x3, x4, …} addition no 4. {multiples of 5}, multiplication no 2 3 2 2 3. , , , … , addition no 12 2. {integers}, multiplication no 1. {integers}, addition yes Tell whether the set forms a group with respect to the given operation. Exercises The set {1, 1} is a group with respect to multiplication because all four properties hold. 1(1) 1; 1(1) 1 The identity for multiplication is 1. Identity: Associative Property: (1)[(1)(1)] (1)(1) 1; and so on The set is associative for multiplication. Is the set {1, 1} a group with respect to multiplication? Closure Property: (1)(1) 1; (1)(1) 1; (1)(1) 1; (1)(1) 1 The set has closure for multiplication. Example 2 Chapter 1 19 Glencoe Algebra 2 Sample answer: When you look at your reflection, you are looking at yourself. The reflexive property says that every number is equal to itself. In geometry, symmetry with respect to a line means that the parts of a figure on the two sides of a line are identical. The symmetric property of equality allows you to interchange the two sides of an equation. The equal sign is like the line of symmetry. 3. How can the words reflection and symmetry help you remember and distinguish between the reflexive and symmetric properties of equality? Think about how these words are used in everyday life or in geometry. Remember What You Learned 2. When Louisa rented a moving truck, she agreed to pay $28 per day plus $0.42 per mile. If she kept the truck for 3 days and the rental charges (without tax) were $153.72, how many miles did Louisa drive the truck? 3(28) 0.42m 153.72 Read the following problem and then write an equation that you could use to solve it. Do not actually solve the equation. In your equation, let m be the number of miles driven. Sample answer: An equation is a statement that says that two algebraic expressions are equal. c. How are algebraic expressions and equations related? Sample answer: Equations contain equal signs; expressions do not. b. How are algebraic expressions and equations different? Sample answer: Both contain variables, constants, and operation signs. 1. a. How are algebraic expressions and equations alike? Read the Lesson Answers The set is not a group with respect to addition because only three of the four properties hold. Is there some number, i, in the set such that i a a a i for all a? 0 1 1 1 0; 0 2 2 2 0; and so on. The identity for addition is 0. (220 A) I P or P (220 A) I 6 6 • Write an equation that shows how to calculate your target heart rate. and desired intensity level (I ) • To find your target heart rate, what two pieces of information must you supply? age (A) Read the introduction to Lesson 1-3 in your textbook. ____________ PERIOD _____ 8:03 AM Identity: Associative Property: For all numbers in the set, does a (b c) (a b) c? 0 (1 2) (0 1) 2; 1 (2 3) (1 2) 3; and so on. The set is associative for addition. Does the set {0, 1, 2, 3, …} form a group with respect to addition? Closure Property: For all numbers in the set, is a b in the set? 0 1 1, and 1 is in the set; 0 2 2, and 2 is in the set; and so on. The set has closure for addition. Solving Equations Lesson Reading Guide Get Ready for the Lesson 1-3 NAME ______________________________________________ DATE 5/19/06 Example 1 A set of numbers forms a group with respect to an operation if for that operation the set has (1) the Closure Property, (2) the Associative Property, (3) a member which is an identity, and (4) an inverse for each member of the set. ____________ PERIOD _____ A2-01-873971 Properties of a Group 1-2 NAME ______________________________________________ DATE Lesson 1-3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A8 (Lessons 1-2 and 1-3) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Solving Equations Study Guide and Intervention ____________ PERIOD _____ division times, product, of (as in of a number) divided by, quotient 25 A9 n Chapter 1 5n 11. n 8 n3 20 Glencoe Algebra 2 The quotient of five times a number and the sum of the number and 3 is equal to the difference of the number and 8. square of the number is equal to four times the number. 10. 2(n3 3n2) 4n Twice the sum of the cube of a number and three times the 9. 3n 35 79 The difference of three times a number and 35 is equal to 79. Write a verbal sentence to represent each equation. Sample answers are given. 8. 23 more than the product of 7 and a number 7n 23 7. four times the square of a number increased by five times the same number 4n 2 5n 6. the product of 3 and the sum of 11 and a number 3(11 n) 5. the sum of 100 and four times a number 100 4n 4. the difference of nine times a number and the quotient of 6 and the same number 3. 7 less than fifteen times a number 15n 7 140 140 100 40 5 Solve 100 8x 140. c 1 3 2 4 4 5 k 2 d 2 20. 2xy x 7, for x x Answers Glencoe Algebra 2 21 10 3 Chapter 1 4 3 25. 4x 3y 10, for y y x 20n 5n 1 23. 3.5s 42 14t, for s s 4t 12 24. 5m 20, for m m m n 22. 3(2j k) 108, for j j 18 f 4 4r q 21. 6, for f f 24 2d 3pq r 7 2y 1 s 20 19. 12, for p p s 2t 17. 10, for t t 18. h 12g 1, for g g h1 12 16. a 3b c, for b b ac 3 Solve each equation or formula for the specified variable. 15. 2x 75 102 x 9 14. 100 20 5r 16 13. 4n 20 53 2n 5 9. 5(4 k) 10k 4 6. 8 2(z 7) 3 3. 5t 1 6t 5 4 y 20 x 12. 4.5 2p 8.7 2.1 1 2 1 5 y (100 4x) 11. n 98 n 28 5 2 8. 3x 17 5x 13 15 5. 7 x 3 8 2. 17 9 a 8 1 2 Solve 4x 5y 100 for y. 10. 120 y 60 80 3 4 7. 0.2b 10 50 2 3 4. m 1. 3s 45 15 Example 2 4x 5y 100 4x 5y 4x 100 4x 5y 100 4x Solve each equation. Check your solution. Exercises 100 8x 100 8x 100 8x x Example 1 c For any real numbers a, b, and c, if a b, a b then a c b c and, if c is not zero, . Multiplication and Division Properties of Equality Answers 2. four times the sum of a number and 3 4(n 3) 1. the sum of six times a number and 25 6n 6 9n Six times the difference of a number and two is equal to 14. Write an algebraic expression to represent each verbal expression. Exercises n 18 3 Example 1 Write an algebraic expression to represent 18 less than the quotient of a number and 3. Example 2 Write a verbal sentence to represent 6(n 2) 14. multiplication minus, difference, decreased by, less than For any real numbers a, b, and c, if a b, then a c b c and a c b c. Addition and Subtraction Properties of Equality 8:03 AM 1 2 addition subtraction and, plus, sum, increased by, more than Operation Word Expression multiplication, or division. (continued) You can solve equations by using addition, subtraction, Solving Equations Study Guide and Intervention ____________ PERIOD _____ 5/19/06 Properties of Equality 1-3 NAME ______________________________________________ DATE A2-01-873971 Verbal Expressions to Algebraic Expressions The chart suggests some ways to help you translate word expressions into algebraic expressions. Any letter can be used to represent a number that is not known. 1-3 NAME ______________________________________________ DATE Lesson 1-3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A9 (Lesson 1-3) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Solving Equations Skills Practice ____________ PERIOD _____ A10 y 3 A number divided by 3 is the difference of 2 and twice the number. 10. 2 2y 22. 2.2n 0.8n 5 4n 5 2a 21. a 3 5 Chapter 1 22 Glencoe Algebra 2 A 2 r 2 2 r 26. A 2r2 2rh, for h h xy 25. A , for y 2 y 2A x 1 24. y x 12, for x 4 I 23. I prt, for p p rt Solve each equation or formula for the specified variable. x 4y 48 20. 4v 20 6 34 5 19. 5x 3x 24 3 5 18. 3b 7 15 2b 17. 3t 2t 5 5 22 5 16. x 4 5x 2 15. 4m 2 18 4 1 2 Substitution () 14. If (8 7)r 30, then 15r 30. Subtraction () 12. If d 1 f, then d f 1. Solve each equation. Check your solution. Symmetric () 13. If 7x 14, then 14 7x. Transitive () 11. If a 0.5b, and 0.5b 10, then a 10. are given. The sum of 8 and 3 times a number is 5. 8. 8 3x 5 are given. of a number and 5 is 3 times the sum of twice the number and 1. 8. 3(2m 1) The quotient m 5 Three times a number is 4 times the cube of the number. 6. 3y 4y3 1 2 5 1 8 4 1 6 3c 1 2 2 Chapter 1 23 Glencoe Algebra 2 26. GOLF Luis and three friends went golfing. Two of the friends rented clubs for $6 each. The total cost of the rented clubs and the green fees for each person was $76. What was the cost of the green fees for each person? g green fees per person; 6(2) 4g 76; $16 25. GEOMETRY The length of a rectangle is twice the width. Find the width if the perimeter is 60 centimeters. w width; 2(2w) 2w 60; 10 cm 2(E U ) w 1 24. E Iw2 U, for I I 2 2d 1 3 22. c , for d d Define a variable, write an equation, and solve the problem. E c h gt 2 23. h vt gt2, for v v t 21. E mc2, for m m 2 Solve each equation or formula for the specified variable. 20. 6y 5 3(2y 1) 4 5 19. 5(6 4v) v 21 3 7 11 1 12 5 18. 6x 5 7 9x 3 4 16. s 5 6 14. 9 4n 59 17 Multiplication () 12. If 4m 15, then 12m 45. Division () 10. If 8(2q 1) 4, then 2(2q 1) 1. 17. 1.6r 5 7.8 8 15. n 3 4 13. 14 8 6r 1 Solve each equation. Check your solution. Subtraction () 11. If h 12 22, then h 10. Symmetric () 9. If t 13 52, then 52 t 13. Name the property illustrated by each statement. Three times a number is twice the difference of the number and 1. 7. 3c 2(c 1) The difference of 5 and twice a number is 4. 5. 5 2x 4 Answers Name the property illustrated by each statement. Three added to the square of a number is the number. 9. b2 3 b The difference of a number and 8 is 16. 7. n 8 16 2y 2 1 4. 1 less than twice the square of a number n (n 1) 2. the sum of two consecutive integers Write a verbal expression to represent each equation. 5–8. Sample answers 5(m 1) 3. 5 times the sum of a number and 1 y 2 5 1. 2 more than the quotient of a number and 5 Write an algebraic expression to represent each verbal expression. 8:03 AM Write a verbal expression to represent each equation. 7–10. Sample answers 6. the product of 11 and the square of a number 11n2 5. 3 times the difference of 4 and a number 3(4 n) 3n 9 4. the product of 3 and a number, divided by 9 5n 8 2. 8 less than 5 times a number Solving Equations Practice ____________ PERIOD _____ 5/19/06 6(n 5) 3. 6 times the sum of a number and 5 4n 7 1. 4 times a number, increased by 7 1-3 NAME ______________________________________________ DATE A2-01-873971 Write an algebraic expression to represent each verbal expression. 1-3 NAME ______________________________________________ DATE Lesson 1-3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A10 (Lesson 1-3) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Solving Equations A11 Chapter 1 w 6 units w w 10 3. GEOMETRY The length of a rectangle is 10 units longer than its width. If the total perimeter of the rectangle is 44 units, what is the width? 24 3t 80 dominoes Glencoe Algebra 2 6. How many dominoes did Nancy have in her hallway? 321 4N 1. 5. Nancy measures her dominoes and finds that t 1 centimeter. She measures the distance of her hallway and finds that d 321 centimeters. Rewrite the equation that relates d, t, and N with the given values substituted for t and d. t Chapter 1 25 Answers Glencoe Algebra 2 There is a surplus. Consumption is reduced, possibly because the cost of living has reduced. 4. Determine if there is a trade surplus or deficit when there is 12 trillion dollar GNP, 2 trillion in investments, 3 trillion in government investments, and 5 trillion in consumption. Explain why this situation may be favorable. Consumption will increase. 3. If the GNP remains steady, and so do investments and government spending, but the trade deficit increases (to say 2 or 3 trillion dollars), what does this say about the consumption level? Government spending must have gone up, in fact G 6 trillion dollars. Employment may have caused a dip in consumption. 2. In 2001, the U.S. trade deficit remain at 1 trillion dollars, investments also remain steady at 1 trillion dollars. However consumption dipped to only 50% of the GNP, which increased to 12 trillion dollars. What was the effect on government spending? What might have caused the change? 10.5 6.3 1 G 1. Therefore G 4.2 trillion dollars. 1. The most important sector of the U.S. economy is consumption. It makes up about 60% of the entire GNP. In 2000, the U.S.’s GNP was 10.5 trillion dollars. In the same year, there were 1 trillion dollars in investments, but a 1 trillion dollar trade deficit. Assuming that consumption made up 60% of the GNP, how much did the government budget for spending? Answers 20 suitcases 2. AIRPLANES The number of passengers p and the number of suitcases s that an airplane can carry are related by the equation 180p + 60s = 3,000. If 10 people board the aircraft, how many suitcases can the airplane carry? Nancy is setting up a train of dominos from the front entrance straight down the hall to the kitchen entrance. The thickness of each domino is t. Nancy places the dominoes so that that the space separating consecutive dominoes is 3t. The total distance that N dominoes takes up is given by d = t(4N + 1). the information below. C is consumer goods (e.g. TV’s, Cars, Food, Furniture, Clothes, Doctors’ fees, and Dining) I is investments (e.g. Factories, Computers, Airlines, and Housing) G is government spending and investments (e.g. Ships, Roads, Schools, NASA, and Bombs) X is exports (e.g. Corn, Wheat, Cars, and Computers) M is for imports, (e.g. Cars, Computer chips, Clothes, and Oil) Calculated from GNP C I G X M, where 8:03 AM DOMINOES For Exercises 5 and 6, use 15 United States’ Gross National Product Enrichment ____________ PERIOD _____ The Gross National Product, GNP, is an important indicator of U.S. economy. The GNP contains information about the inflation rate, the Bond market, and the Stock market. It is composed of consumer goods, investments, government expenditures, exports, and imports. 1-3 NAME ______________________________________________ DATE 5/19/06 F 3R 5. 4. SAVINGS Jason started with d dollars in his piggy bank. One week later, Jason doubled the amount in his piggy bank. Another week later, Jason was able to add $20 to his piggy bank. At this point, the piggy bank had $50 in it. What is d? Word Problem Practice ____________ PERIOD _____ A2-01-873971 1. AGES Robert’s father is 5 years older than 3 times Robert’s age. Let Robert’s age be denoted by R and let Robert’s father’s age be denoted by F. Write an equation that relates Robert’s age and his father’s age. 1-3 NAME ______________________________________________ DATE Lesson 1-3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A11 (Lesson 1-3) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Solving Equations and Checking Solutions Graphing Calculator Activity ____________ PERIOD _____ Solve 2(5y 1) y 4(y 3). ENTER 3 ENTER — ) 2 5 2nd — 1 6 ZOOM [QUIT] ZOOM ( 8 ) MATH ENTER — 2nd ENTER (–) [CALC] 5 . ENTER ENTER 4 2 A12 ZOOM ENTER VARS 2nd — 6 ) ( [CALC] 2. ) 2 — ENTER 5 — 2 ) ( ENTER 4 ENTER VARS ) 11 Chapter 1 4 z 1 4. 3(2z 25) 2(z 1) 78 2 w 5 1. 3(2w 7) 9 2(5w 4) Solve each equation. Exercises 26 m 8 3 5 m4 3m 1 5. 1 x 17 2. 1.5(4 x) 1.3(2 x) 2 2 8 11 21 x Glencoe Algebra 2 x5 1 x3 6. 2x 5 a 4 4 3. 1(a 2) 1(5 a) 6 [10, 10] scl:1 by [10, 10] scl:1 20 Use arrow keys and enter to set the bound prompts. The solution is x . ENTER ENTER Keystrokes: Y = ( ( 1 2 ENTER Graph the expression on the left side of the equation in Y1 and the expression on the right side of the equation in Y2. Enter Y1 - Y2 in Y3. Then graph the function in Y3. Use the zero function under the CALC menu to determine where the graph of Y3 equals zero. This point will be the solution. 5 Solve x x 1(x 2). [47, 47] scl:10 by [31, 31] scl:10 Chapter 1 27 Glencoe Algebra 2 Sample answer: The number line shows that for every positive number, there are two numbers that have that number as their absolute value. 5. How can the number line model for absolute value that is shown on page 28 of your textbook help you remember that many absolute value equations have two solutions? Remember What You Learned is , then there are no solutions. 4. What does the symbol mean as a solution set? Sample answer: If a solution set greater than 0. 3. What does the sentence b 0 mean? Sample answer: The number b is 0 or The absolute value is the number of units it is from 0 on the number line. The number of units is never negative. 2. Explain why the absolute value of a number can never be negative. Sample answer: answer: If a is negative, then a is positive. Example: If a 25, then a (25) 25. 1. Explain how a could represent a positive number. Give an example. Sample Read the Lesson • If the magnitude of an earthquake is estimated to be 6.9 on the Richter scale, it might 6.6 or as high as 7.2 . actually have a magnitude as low as from the measured magnitude by up to 0.3 unit in either direction, so an absolute value equation is needed. Answers Example 2 7 10 is the value of both sides of the equation when x . 7 10 The x-coordinate, , is the solution to the equation. The y-coordinate ENTER (–) 2:55 PM Y= • Why is an absolute value equation rather than an equation without absolute value used to find the extremes in the actual magnitude of an earthquake in relation to its measured value on the Richter scale? Sample answer: The actual magnitude can vary who studies earthquakes; a number from 1 to 10 that tells how strong an earthquake is • What is a seismologist and what does magnitude of an earthquake mean? a scientist Read the introduction to Lesson 1-4 in your textbook. 5/25/06 Keystrokes: 4 ( Graph the expression on the left side of the equation in Y1 and the expression on the right side of the equation in Y2. Choose an appropriate view window so that the intersection of the graphs is visible. Then use the intersect command to find the coordinates of the common point. Example 1 Solving Absolute Value Equations Lesson Reading Guide Get Ready for the Lesson 1-4 NAME ______________________________________________ DATE______________ PERIOD _____ A2-01-873971 When solving equations, checking the solutions is an important process. A graphing calculator can be used to check the solution of an equation. 1-3 NAME ______________________________________________ DATE Lesson 1-4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A12 (Lessons 1-3 and 1-4) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Solving Absolute Value Equations Study Guide and Intervention 1 2 ⏐2x 3y⏐ ⏐2(4) 3(3)⏐ ⏐8 9⏐ ⏐17⏐ 17 Evaluate 2x 3y if x 4 and y 3. Example 2 A13 Chapter 1 22. ⏐yz 4w⏐ w 17 1 2 28 23. ⏐wz⏐ ⏐8y⏐ 20 3 4 20. 12 ⏐10x 10y⏐ 3 17. ⏐2x y⏐ 5y 6 16. 14 2⏐w xy⏐ 4 19. z⏐z⏐ x⏐x⏐ 32 14. 3⏐wx⏐ ⏐4x 8y⏐ 27 13. ⏐6y z⏐ ⏐yz⏐ 6 1 4 11. ⏐z⏐ 4⏐2z y⏐ 40 8. ⏐wz⏐ ⏐xy⏐ 23 7. ⏐w 4x⏐ 12 10. 5⏐w⏐ 2⏐z 2y⏐ 34 5. ⏐x⏐ ⏐y⏐ ⏐z⏐ 4 4. ⏐x 5⏐ ⏐2w⏐ 1 1 2 2. ⏐6 z⏐ ⏐7⏐ 7 Glencoe Algebra 2 24. xz ⏐xz⏐ 24 1 2 21. ⏐5z 8w⏐ 31 18. ⏐xyz⏐ ⏐wxz⏐ 54 15. 7⏐yz⏐ 30 9 12. 10 ⏐xw⏐ 2 9. ⏐z⏐ 3⏐5yz⏐ 39 6. ⏐7 x⏐ ⏐3x⏐ 11 3. 5 ⏐w z⏐ 15 a 2x 3 2x 3 3 2x x b 17 17 3 20 10 1 7 b 17 17 3 14 7 11 , 1 2 Chapter 1 1 2 12 15. ⏐6 2x⏐ 3x 1 29 Answers 16. ⏐16 3x⏐ 4x 12 {4} Glencoe Algebra 2 14. ⏐4b 3⏐ 15 2b {2, 9} 13. 5f ⏐3f 4⏐ 20 {12} 10. ⏐3x 1⏐ 2x 11 {2, 12} 8. ⏐8 5a⏐ 14 a 6. ⏐15 2k⏐ 45 {15, 30} 4. ⏐m 3⏐ 12 2m {3} 2. ⏐t 4⏐ 5 0 {1, 9} ⏐14 3⏐ 17 ⏐17⏐ 17 17 17 ✓ 12. 40 4x 2⏐3x 10⏐ {6, 10} ⏐ 11. x 3 1 1 ⏐3 9. ⏐4p 11⏐ p 4 23, 1 3 7. 5n 24 ⏐8 3n⏐ {2} 5. ⏐5b 9⏐ 16 2 3. ⏐x 5⏐ 45 {40, 50} 1. ⏐x 15⏐ 37 {52, 22} Solve each equation. Check your solutions. Exercises a 2x 3 2x 3 3 2x x CHECK ⏐2(7) 3⏐ 17 Case 2 Solve 2x 3 17. Check your solutions. ⏐2x 3⏐ 17 ⏐2(10) 3⏐ 17 ⏐20 3⏐ 17 ⏐17⏐ 17 17 17 ✓ There are two solutions, 10 and 7. CHECK Case 1 Example Answers 1. ⏐2x 8⏐ 4 Evaluate each expression if w 4, x 2, y , and z 6. Exercises ⏐4⏐ ⏐2x⏐ ⏐4⏐ ⏐2 6⏐ ⏐4⏐ ⏐12⏐ 4 12 8 x 6. Evaluate 4 2x if For any real numbers a and b, where b 0, if ⏐a⏐ b then a b or a b. Always check your answers by substituting them into the original equation. Sometimes computed solutions are not actual solutions. 8:03 AM Example 1 Absolute Value For any real number a, if a is positive or zero, the absolute value of a is a. If a is negative, the absolute value of a is the opposite of a. • Symbols For any real number a, ⏐a⏐ a, if a 0, and ⏐a⏐ a, if a 0. • Words (continued) Use the definition of absolute value to solve equations containing absolute value expressions. Absolute Value Equations Solving Absolute Value Equations Study Guide and Intervention 5/19/06 The absolute value of a number is the number of units it is from 0 on a number line. The symbol ⏐x⏐ is used to represent the absolute value of a number x. 1-4 NAME ______________________________________________ DATE______________ PERIOD _____ A2-01-873971 Absolute Value Expressions 1-4 NAME ______________________________________________ DATE______________ PERIOD _____ Lesson 1-4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A13 (Lesson 1-4) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Solving Absolute Value Equations Skills Practice ____________ PERIOD _____ 6. ⏐8x 3y⏐ ⏐2y 5x⏐ 21 8. 44 ⏐2x y⏐ 45 5. ⏐10z 31⏐ 131 7. 25 ⏐5z 1⏐ 24 12. 6.4 ⏐w 1⏐ 7 A14 1 5 2 6 5 1 6 6 Chapter 1 25. 5⏐6a 2⏐ 15 , 1 2 30 26. ⏐k⏐ 10 9 Glencoe Algebra 2 24. ⏐8d 4d⏐ 5 13 {2, 2} 22. 4⏐7 y⏐ 1 11 {4, 10} 21. ⏐7x 3x⏐ 2 18 {4, 4} 23. ⏐3n 2⏐ , 20. 2⏐3w⏐ 12 {2, 2} 19. ⏐p 7⏐ 14 16. ⏐2g 6⏐ 0 {3} 18. ⏐2x x⏐ 9 {3, 3} 17. 10 ⏐1 c⏐ {9, 11} 4 8 15. ⏐3k 6⏐ 2 , 3 3 14. ⏐5a⏐ 10 {2, 2} 12. ⏐2 2d⏐ 3⏐b⏐ 19.2 16. 7⏐x 3⏐ 42 {9, 3} 18. ⏐5x 4⏐ 6 15. ⏐2y 3⏐ 29 {13, 16} 17. ⏐3u 6⏐ 42 {12, 16} 2 3 28. 3 5⏐2d 3⏐ 4 27. 5⏐2r 3⏐ 5 0 {2, 1} Chapter 1 x 51 3 or 48 x 54 31 Glencoe Algebra 2 30. OPINION POLLS Public opinion polls reported in newspapers are usually given with a margin of error. For example, a poll with a margin of error of 5% is considered accurate to within plus or minus 5% of the actual value. A poll with a stated margin of error of 3% predicts that candidate Tonwe will receive 51% of an upcoming vote. Write and solve an equation describing the minimum and maximum percent of the vote that candidate Tonwe is expected to receive. x 87.4 1.5; or 85.9 x 88.9 29. WEATHER A thermometer comes with a guarantee that the stated temperature differs from the actual temperature by no more than 1.5 degrees Fahrenheit. Write and solve an equation to find the minimum and maximum actual temperatures when the thermometer states that the temperature is 87.4 degrees Fahrenheit. 26. 5 3⏐2 2w⏐ 7 {3, 1} 25. 2⏐4 s⏐ 3s {8} 24. 2⏐7 3y⏐ 6 14 1, 3 22. ⏐4w 1⏐ 5w 37 {38} 21. ⏐8 p⏐ 2p 3 {11} 23. 4⏐2y 7⏐ 5 9 {3, 4} 20. 6⏐5 2y⏐ 9 1.75, 3.25 19. 3⏐4x 9⏐ 24 14. ⏐x 13⏐ 2 {11, 15} 13. ⏐n 4⏐ 13 {9, 17} Solve each equation. Check your solutions. 11. ⏐a b⏐ ⏐b a⏐ 14 Answers 13. ⏐y 3⏐ 2 {5, 1} Solve each equation. Check your solutions. 11. ⏐3x 2y⏐ 4 4 8. ⏐1 7c⏐ ⏐a⏐ 33 7. ⏐5a 7⏐ ⏐3c 4⏐ 23 10. ⏐4d⏐ ⏐5 2a⏐ 12.6 6. ⏐2b 1⏐ ⏐8b 5⏐ 52 5. 6⏐10a 12⏐ 132 9. 3⏐0.5c 2⏐ ⏐0.5b⏐ 17.5 4. ⏐17c⏐ ⏐3b 5⏐ 114 3. ⏐10d a⏐ 15 8:03 AM 10. 3 ⏐1 6w⏐ 1.6 4. ⏐17z⏐ 170 3. ⏐9y z⏐ 17 2. ⏐2b 4⏐ 12 1. ⏐6a⏐ 6 5/19/06 9. 2⏐4w⏐ 3.2 2. ⏐9y⏐ 27 1. ⏐5w⏐ 2 Solving Absolute Value Equations Practice Evaluate each expression if a 1, b 8, c 5, and d 1.4. 1-4 NAME ______________________________________________ DATE______________ PERIOD _____ A2-01-873971 Evaluate each expression if w 0.4, x 2, y 3, and z 10. 1-4 NAME ______________________________________________ DATE Lesson 1-4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A14 (Lesson 1-4) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 12 14 16 18 20 22 24 A15 Chapter 1 |a 200| 3 a 197 or 203 32 Glencoe Algebra 2 They will be equal only if Jim walks in the same direction each time giving a, b, and c all the same sign. 7. When would the formula you wrote in part A give the same value as the formula shown in part B? The distance Jim ends up from where he started. 6. The equation you wrote in part A should not be T |a b c|. What does |a b c| represent? T |a| |b | |c | 5. Write a formula for the total distance that Jim walked. Jim is walking along a straight line. An observer watches him. If Jim walks forward, the observer records the distance as a positive number, but if he walks backward, the observer records the distance as a negative number. The observer has recorded that Jim has walked a, then b, then c feet. WALKING For Exercises 5–7, use the following information. x 2 3 (x 6) (x 2) 3 x 6 4. | x 4 | 6 | x 3 | x 2.5 3. |3x 6 | |5x 10 | x 2 Chapter 1 Answers (x 2) (2x 4) (x 3) Glencoe Algebra 2 x 2 (2x 4) x 3 x 2 (2x 4) (x 3) (x 2) (2x 4) x 3 33 (x 2) 2x 4 (x 3) x 2 2x 4 (x 3) no solution (x 2) 2x 4 x 3 x 2 2x 4 x 3 6. List each case and solve | x 2 | | 2x 4 | | x 3 |. Check your solution. 5. How many cases would there be for an absolute value equation containing three sets of absolute value symbols? 8 2. |2x 9 | | x 3 | x 12, 2 1. | x 4 | | x 7 | x 1.5 Solve each absolute value equation. Check your solution. Exercises | x 2 | 3 | x 6 |. The only solution to this equation is 52. Solve each of these equations and check your solutions in the original equation, x 2 3 (x 6) x23x6 Each of these equations also has two cases. By writing the equations for both cases of each equation above, you end up with the following four equations: Answers 3. AGES Rhonda conducts a survey of the ages of students in eleventh grade at her school. On November 1, she finds the average age is 200 months. She also finds that two-thirds of the students are within 3 months of the average age. Write and solve an equation to determine the age limits for this group of students. d |s j| or d |j s| 2. HEIGHT Sarah and Jessica are sisters. Sarah’s height is s inches and Jessica’s height is j inches. Their father wants to know how many inches separate the two. Write an equation for this difference in such a way that the result will always be positive no matter which sister is taller. 13 or 21 10 | x 2| 3 x 6 | x 2 | 3 (x 6) Consider the problem | x 2 | 3 | x 6 |. First we must write the equations for the case where x 6 0 and where x 6 0. Here are the equations for these two cases: You have learned that absolute value equations with one set of absolute value symbols have two cases that must be considered. For example, | x 3 | 5 must be broken into x 3 5 or (x 3) 5. For an equation with two sets of absolute value symbols, four cases must be considered. 8:03 AM Maria’s | w 10| 0.1 minimum weight: 9.9 pounds maximum weight: 10.1 pounds Enrichment ____________ PERIOD _____ Considering All Cases in Absolute Value Equations 1-4 NAME ______________________________________________ DATE 5/19/06 4. TOLERANCE Martin makes exercise weights. For his 10 pound dumbbells, he guarantees that the actual weight of his dumbbells is within 0.1 pounds of 10 pounds. Write and solve an equation that describes the minimum and maximum weight of his 10 pound dumbbells. Solving Absolute Value Equations Word Problem Practice ____________ PERIOD _____ A2-01-873971 1. LOCATIONS Identical vacation cottages, equally spaced along a street, are numbered consecutively beginning with 10. Maria lives in cottage #17. Joshua lives 4 cottages away from Maria. If n represents Joshua’s cottage number, then |n 17| 4. What are the possible numbers of Joshua’s cottage? 1-4 NAME ______________________________________________ DATE Lesson 1-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A15 (Lesson 1-4) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Absolute Value Statements Spreadsheet Activity A16 Chapter 1 34 5. If a and b are real numbers, then c|a b| c|a| |b|. sometimes 4. If a and b are real numbers, then |a| |b| a b. sometimes 3. If a and b are real numbers, then |a b| x. never 2. If a and b are real numbers, then |a b| |a| |b|. sometimes 1. For all real numbers a and b, a 0, |ax b| 0. sometimes Use a spreadsheet to determine whether each absolute value statement is sometimes, always, or never true. Exercises Through observation of Column D, when c is negative the statement is not true. The absolute value statement, c|a b| |ca cb| is sometimes true; it is true only if c 0. Plan 2: $55 2 3 4 5 4 3 2 1 0 5 4 3 2 1 0 ⫺1 ⫺0 1 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 1 5 1 2 2 6 2 3 3 7 3 4 4 8 4 5 5 9 5 {x x 1} {x x 2} {x x 5} {x x, 3} Chapter 1 35 Glencoe Algebra 2 numbers, for example, 3 and 8. Then plot their additive inverses, 3 and 8. Write an inequality that compares the positive numbers and one that compares the negative numbers. Notice that 8 3, but 8 3. The order changes when you multiply by 1. 3. One way to remember something is to explain it to another person. A common student error in solving inequalities is forgetting to reverse the inequality symbol when multiplying or dividing both sides of an inequality by a negative number. Suppose that your classmate is having trouble remembering this rule. How could you explain this rule to your classmate? Sample answer: Draw a number line. Plot two positive Remember What You Learned d. There is space for at most 165 students in the high school band. 165 c. To participate in a concert, you must be willing to attend at least ten rehearsals. 10 b. A student may enroll in no more than six courses each semester. 6 a. There are fewer than 600 students in the senior class. 600 2. Show how you can write an inequality symbol followed by a number to describe each of the following situations. d. c. b. a. 1. There are several different ways to write or show inequalities. Write each of the following in interval notation. Read the Lesson Plan 2 Answers Step 2 Use Column D to test the equation. A formula such as C2*ABS(A2B2) ABS(C2*A2C2*B2) in cell D2 returns TRUE if the equation is true. $65 Which plan should you choose? Plan 1: • Suppose that in one month you use 475 minutes of airtime on your wireless phone. Find your monthly cost with each plan. • Write an inequality comparing the number of minutes per month included in the two phone plans. 150 400 or 400 150 Read the introduction to Lesson 1-5 in your textbook. 8:03 AM Step 1 Use Columns A, B, and C for the values of a, b, and c. Choose several sets of values including positive and negative numbers, and zero. Solving Inequalities Lesson Reading Guide Get Ready for the Lesson 1-5 NAME ______________________________________________ DATE______________ PERIOD _____ 5/19/06 Determine whether c|a b| |ca cb| is sometimes, always, or never true. Try a number of values for a, b, and c to determine whether the statement is true or false for each set of values. Glencoe Algebra 2 ____________ PERIOD _____ A2-01-873971 You can use a spreadsheet to try several different values in an equation to help you determine whether the statement is sometimes, always, or never true. Remember that showing that a statement is true for some values does not prove that it is true for all values. However, finding one value for which a statement is false proves that it is not true for all values. 1-4 NAME ______________________________________________ DATE Lesson 1-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A16 (Lessons 1-4 and 1-5) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 A17 1 2 Chapter 1 4 3 2 1 0 1 xx 2 1 2 3 7. 4x 2 7(4x 2) 3 4 4 8 7 6 5 4 3 2 1 0 {k k 4} 4. 18 4k 2(k 21) 4 3 2 1 0 {a a 3} 1. 7(7a 9) 84 9 8 7 6 5 4 3 2 1 17 3w 35 17 3w 17 35 17 3w 18 w 6 The solution set is {w⏐w 6}. 1 1 3 8 12 36 10 {y y 9} 14 3 4 8 6 9 10 11 12 13 14 2 8. (2y 3) y 2 7 {b b 11} 5. 4(b 7) 6 22 4 3 2 1 0 {z z 2} 2. 3(9z 4) 35z 4 6 c c c Solve 17 3w 35. Then graph the solution set on a number line. Example 2 1 2 3 4 5 6 7 8 Glencoe Algebra 2 19 20 21 22 23 24 25 26 27 {d d 24} 9. 2.5d 15 75 0 {m m 5} 6. 2 3(m 5) 4(m 3) 8 7 6 5 4 3 2 1 0 {n n 7} 3. 5(12 3n) 165 Solve each inequality. Describe the solution set using set-builder notation. Then graph the solution set on a number line. Exercises c a b 4. If c is negative and a b, then ac bc and . is at least is no less than is greater than or equal to Chapter 1 4 strings. 37 Answers Glencoe Algebra 2 5. Solve the inequality and interpret the solution. The friends can bowl at most 4. Write an equation to describe this situation. 3(3.50) 4(1.50)n 40 Four friends plan to spend Friday evening at the bowling alley. Three of the friends need to rent shoes for $3.50 per person. A string (game) of bowling costs $1.50 per person. If the friends pool their $40, how many strings can they afford to bowl? BOWLING For Exercises 4 and 5, use the following information. per hour in order to finish the book in less than 6 hours. 3. Solve the inequality and interpret the solution. Ethan must read at least 67 pages 2. Write an inequality to describe this situation. 6 or 6n 482 80 482 80 n Ethan is reading a 482-page book for a book report due on Monday. He has already read 80 pages. He wants to figure out how many pages per hour he needs to read in order to finish the book in less than 6 hours. PLANNING For Exercises 2 and 3, use the following information. At most 17 hours 1. PARKING FEES The city parking lot charges $2.50 for the first hour and $0.25 for each additional hour. If the most you want to pay for parking is $6.50, solve the inequality 2.50 0.25(x 1) 6.50 to determine for how many hours you can park your car. Exercises Let x be the number of remaining games that the Vikings must win. The total number of games they will have won by the end of the season is 16 x. They want to win at least 80% of their games. Write an inequality with . 16 x 0.8(36) x 0.8(36) 16 x 12.8 Since they cannot win a fractional part of a game, the Vikings must win at least 13 of the games remaining. Example SPORTS The Vikings play 36 games this year. At midseason, they have won 16 games. How many of the remaining games must they win in order to win at least 80% of all their games this season? is at most is no more than is less than or equal to Answers 13 14 15 16 17 18 19 20 21 2x 4 4 36 4 2x 32 x 16 The solution set is {x⏐x 16}. Solve 2x 4 36. Then graph the solution set on a number line. Example 1 These properties are also true for and . c a b 3. If c is negative and a b, then ac bc and . is greater than is more than is less than is fewer than 8:03 AM c a b 2. If c is positive and a b, then ac bc and . c For any real numbers a, b, and c, with c 0: a b 1. If c is positive and a b, then ac bc and . For any real numbers a, b, and c: 1. If a b, then a c b c and a c b c. 2. If a b, then a c b c and a c b c. c Multiplication and Division Properties for Inequalities Addition and Subtraction Properties for Inequalities (continued) Many real-world problems involve inequalities. The chart below shows some common phrases that indicate inequalities. Solving Inequalities Study Guide and Intervention Real-World Problems with Inequalities 1-5 5/19/06 The following properties can be used to solve inequalities. Solving Inequalities Study Guide and Intervention NAME ______________________________________________ DATE______________ PERIOD _____ A2-01-873971 Solve Inequalities 1-5 NAME ______________________________________________ DATE______________ PERIOD _____ Lesson 1-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A17 (Lesson 1-5) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Solving Inequalities Skills Practice z 4 1 2 3 4 5 6 7 1 2 3 4 1 2 3 4 5 6 A18 1 2 3 4 1 2 3 4 5 6 1 2 3 4 5 6 4 1 2 3 4 1 2 3 4 5 6 1 2 3 4 1 2 3 4 3 4 1 2 3 4 1 0 1 2 3 4 5 6 7 14. 4x 9 2x 1 {x x 5} 4 3 2 1 0 12. 4(5x 7) 13 x x or 4 3 2 1 0 7 10. 7t (t 4) 25 t t 2 4 3 2 1 0 8. 7f 9 3f 1 {f f 2} 2 1 0 6. 4b 9 7 {b b 4} 4 3 2 1 0 Chapter 1 6n 5 2n; n 5 4 38 Glencoe Algebra 2 18. Five less than the product of 6 and a number is no more than twice that same number. 1 2 17. One half of a number is more than 6 less than the same number. n n 6; n 12 16. The difference of three times a number and 16 is at least 8. 3n 16 8; n 8 15. Nineteen more than a number is less than 42. n 19 42; n 23 Define a variable and write an inequality for each problem. Then solve. 2 1 0 13. 1.7y 0.78 5 {y y 3.4} 2 1 0 11. 0.7m 0.3m 2m 4 {m m 4} 4 3 2 1 0 3 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 5 6 1 2 3 4 1 2 3 4 1 2 3 4 5 6 1 3 4 5 4 1 2 3 4 1 2 3 4 1 2 3 4 5 7 8 4 3 2 1 0 1 {n n 4} 1 2 3 4 2 3 4 5 6 14. 4n 5(n 3) 3(n 1) 4 4 3 2 1 0 6 12. q 2(2 q) 0 q q 0 Chapter 1 39 Glencoe Algebra 2 18. BANKING Jan’s account balance is $3800. Of this, $750 is for rent. Jan wants to keep a balance of at least $500. Write and solve an inequality describing how much she can withdraw and still meet these conditions. 3800 750 w 500; w $2550 17. HOTELS The Lincoln’s hotel room costs $90 a night. An additional 10% tax is added. Hotel parking is $12 per day. The Lincoln’s expect to spend $30 in tips during their stay. Solve the inequality 90x 90(0.1)x 12x 30 600 to find how many nights the Lincoln’s can stay at the hotel without exceeding total hotel costs of $600. 5 nights 4[2n (3)] 5.5n; n 4.8 16. Four times the sum of twice a number and 3 is less than 5.5 times that same number. 15. Twenty less than a number is more than twice the same number. n 20 2n; n 20 10. 1 5(x 8) 2 (x 5) {x x 6} 4 3 2 1 0 8. 17.5 19 2.5x {x x 0.6} 4 3 2 1 0 2 6. 3(4w 1) 18 w w 4 3 2 1 0 4. 14s 9s 5 {s s 1} 2 1 0 Define a variable and write an inequality for each problem. Then solve. 4 3 2 1 0 {w w 3} 13. 36 2(w 77) 4(2w 52) 4 3 2 1 0 11. 3.5 {x x 1} 2 4x 3 2 1 0 9. 9(2r 5) 3 7r 4 {r r 4} 4 3 2 1 0 7. 1 8u 3u 10 {u u 1} 4 3 2 1 0 2 5. 9x 11 6x 9 x x 3 4 3 2 1 0 3. 16 8r 0 {r r 2} 4 3 2 1 0 2. 23 4u 11 {u u 3} Answers 9. 3s 8 5s {s s 1} 2 1 0 7. 2z 9 5z {z z 3} 4 3 2 1 0 2 4. 20 3s 7s {s s 2} 1 1. 8x 6 10 {x x 2} 8:03 AM 5. 3x 9 {x x 3} 1 0 3. 16 3q 4 {q q 4} 4 3 2 1 0 2. 3a 7 16 {a a 3} 5/19/06 9 8 7 6 5 4 3 2 1 1. 2 {z z 8} Solving Inequalities Practice Solve each inequality. Describe the solution set using set-builder or interval notation. Then, graph the solution set on a number line. 1-5 NAME ______________________________________________ DATE______________ PERIOD _____ A2-01-873971 Solve each inequality. Describe the solution set using set-builder or interval notation. Then, graph the solution set on a number line. 1-5 NAME ______________________________________________ DATE______________ PERIOD _____ Lesson 1-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A18 (Lesson 1-5) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Solving Inequalities A19 Chapter 1 15P 300 1500 P 80; Manuel must translate at least 80 pages. 40 125 pounds Glencoe Algebra 2 7. Ron and his father want to go on the ride together. Ron’s father weighs 175 pounds. What is the maximum weight Ron can be for the two to be allowed on the ride? a 150 6. Write an inequality that describes the limit on the average weight a of the two riders. x y 300 5. Write an inequality that describes the weight limitation in terms of x and y. On a Ferris wheel at a carnival, only two people per car are allowed. The two people together cannot weigh more than 300 pounds. Let x and y be the weights of the people. CARNIVALS For Exercises 5–7, use the following information. It is incorrect. From step 2 to step 3, Brandon must change the direction of the inequality because he is dividing by a negative number. The correct answer is x 6. For all elements a, b, and c of set A, if a R b and b R c, then a R c. Transitive Property , {all lines in a plane} no; reflexive Chapter 1 41 Answers 14. is the greatest integer not greater than, {all integers} yes no; reflexive, symmetric, transitive 13. is the greatest integer not greater than, {all numbers} Glencoe Algebra 2 12. has the same color eyes as, {all members of the Cleveland Symphony Orchestra} yes 11. 10. is the square of, {all numbers} no; reflexive, symmetric, transitive 9. is a multiple of, {all integers} no; symmetric 8. ⊥, {all lines in a plane} no; reflexive, transitive 7. is the spouse of, {all people in Roanoke, Virginia} no; reflexive, transitive 6. , {all polygons in a plane} yes 5. is a factor of, {all nonzero integers} no; symmetric 4. , {all numbers} no; symmetric 3. is the sister of, {all women in Tennessee} no; reflexive 2. , {all triangles in a plane} yes 1. , {all numbers} no; reflexive, symmetric In each of the following, a relation and a set are given. Write yes if the relation is an equivalence relation on the given set. If it is not, tell which of the properties it fails to exhibit. Answers 3. INCOME Manuel takes a job translating English instruction manuals to Spanish. He will receive $15 per page plus $100 per month. Manuel would like to work for 3 months during the summer and make at least $1,500. Write and solve an inequality to find the minimum number of pages Manuel must translate in order to reach his goal. P D 7 2. PARTY FAVORS Janelle would like to give a party bag to every person who is coming to her party. The cost of the party bag is $7 per person. Write an inequality that describes the number of people P that she can invite if Janelle has D dollars to spend on the party bags. For all elements a and b of set A, if a R b, then b R a. Symmetric Property Equality on the set of all real numbers is reflexive, symmetric, and transitive. Therefore, it is an equivalence relation. For any element a of set A, a R a. Reflexive Property A relation R on a set A is an equivalence relation if it has the following properties. 8:03 AM 5 < —2x — 7 12 < —2x —6 < x Enrichment Equivalence Relations 1-5 ____________ PERIOD _____ 5/19/06 b 20 4. FINDING THE ERROR The sample below shows how Brandon solved 5 2x 7. Study his solution and determine if it is correct. Explain your reasoning. Word Problem Practice NAME ______________________________________________ DATE A2-01-873971 1. PANDAS An adult panda bear will eat at least 20 pounds of bamboo every day. Write an inequality that expresses this situation. 1-5 NAME ______________________________________________ DATE______________ PERIOD _____ Lesson 1-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A19 (Lesson 1-5) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 Solving Compound and Absolute Value Inequalities Lesson Reading Guide 11:30 P.M. 5:00 P.M. Jason and Juanita 1 2 3 4 5 A20 2 3 4 5 {x x 1 or x 3} Chapter 1 42 Glencoe Algebra 2 quantity is followed by a or symbol, the expression inside the absolute value bars must be between two numbers, so this becomes an and inequality. The number line graph will show a single interval between two numbers. If the absolute value quantity is followed by a or symbol, it becomes an or inequality, and the graph will show two disconnected intervals with arrows going in opposite directions. 5. Many students have trouble knowing whether an absolute value inequality should be translated into an and or an or compound inequality. Describe a way to remember which of these applies to an absolute value inequality. Also describe how to recognize the difference from a number line graph. Sample answer: If the absolute value Remember What You Learned 4. Write a statement equivalent to ⏐3x 7⏐ 8 that does not use the absolute value symbol. 8 3x 7 8 3. Write a statement equivalent to ⏐4x 5⏐ 2 that does not use the absolute value symbol. 4x 5 2 or 4x 5 2 1 5 4 3 2 1 0 1 2 3 3 4 4 5 5 2 4 and 6 8 2x 5 19 2x 14 x7 8 6 4 2 0 3y 2 7 3y 9 y3 2 or or or 4 2 4 5 7 8 9 11 13 15 17 19 6 Chapter 1 0 2 4 6 8 10 12 14 16 {w 4 w 14} 7. 22 6w 2 82 0 10 20 30 40 50 60 70 80 {y 29 y 54} 5. 100 5y 45 225 3 {x 7 x 15} 3. 18 4x 10 50 8 6 4 2 0 2 3 1 3 4 2 3 4 12 0 12 24 43 4 3 2 1 0 1 2 3 {all real numbers} 4 Glencoe Algebra 2 8. 4d 1 9 or 2d 5 11 24 {b b 12 or b 18} 6. b 2 10 or b 5 4 4 3 2 1 0 {k k 3 or k 2.5} 4. 5k 2 13 or 8k 1 19 10 0 10 20 30 40 50 60 70 {a a 5 or a 52} {x 4 x 4} 1 4 2. 3a 8 23 or a 6 7 8 1. 10 3x 2 14 6 2y 1 9 2y 8 y 4 Example 2 Solve 3y 2 7 or 2y 1 9. Graph the solution set on a number line. The graph is the union of solution sets of two inequalities. Solve each inequality. Graph the solution set on a number line. Exercises 8 6 4 2 0 4 x 7 3 2x 5 8 2x 4 x Example 1 Solve 3 2x 5 19. Graph the solution set on a number line. Or Compound Inequalities 2 The graph is the intersection of solution sets of two inequalities. Answers 5 4 3 2 1 0 2. Use a compound inequality and set-builder notation to describe the following graph. 5 4 3 2 1 0 b. Graph the inequality that you wrote in part a on a number line. 1. a. Write a compound inequality that says, “x is greater than 3 and x is less than or equal to 4.” 3 x 4 Juanita Samir 1 Example: x 3 or x 1 5 4 3 2 1 0 Example: x 4 and x 3 8:03 AM Read the Lesson 5:00 A.M. 1:30 A.M. And Compound Inequalities Compound Inequalities A compound inequality consists of two inequalities joined by the word and or the word or. To solve a compound inequality, you must solve each part separately. Solving Compound and Absolute Value Inequalities Study Guide and Intervention 5/19/06 Ora Jason • Five patients arrive at a medical laboratory at 11:30 A.M. for a glucose tolerance test. Each of them is asked when they last had something to eat or drink. Some of the patients are given the test and others are told that they must come back another day. Each of the patients is listed below with the times when they started to fast. (The P.M. times refer to the night before.) Which of the patients were accepted for the test? Read the introduction to Lesson 1-6 in your textbook. 1-6 NAME ______________________________________________ DATE______________ PERIOD _____ A2-01-873971 Get Ready for the Lesson 1-6 NAME ______________________________________________ DATE______________ PERIOD _____ Lesson 1-6 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A20 (Lesson 1-6) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 (continued) Solving Compound and Absolute Value Inequalities Study Guide and Intervention 4 6 8 8 6 4 2 0 2 4 6 8 4 3 A21 ⏐ 1 2 3 4 8 12 16 20 24 2 4 6 8 40 20 0 20 40 4. ⏐a 9⏐ 30 {a a 39 or a 21} 8 6 4 2 0 2. ⏐4s⏐ 1 27 {s s 6.5 or s 6.5} 2 4 6 8 10 12 1 2 3 4 5 6 7 3 2 8 Chapter 1 9. ⏐4b 11⏐ 17 b b 7 0 7. ⏐10 2k⏐ 2 {k 4 k 6} 4 2 0 ⏐ ⏐ 2 4 6 8 44 40 5 10 15 20 25 30 2 3 Glencoe Algebra 2 10. ⏐100 3m⏐ 20 m m 26 or m 10 5 0 8. 5 2 10 {x x 6 or x 26} x 2 8 6 4 2 0 5. ⏐2f 11⏐ 9 {f f 1 or f 10} 6. ⏐5w 2⏐ 28 {w 6 w 5.2} 8 4 0 3. 3 5 {c 4 c 16} c ⏐2 5 4 3 2 1 0 2 3 4 1 2 3 4 4 3 2 1 0 4 3 2 1 0 1 1 2 2 3 3 4 4 n 3 n 1 4 6 8 6 4 2 0 8. 6. 4 3 2 1 0 4 3 2 1 0 1 1 2 2 2 4 3 3 6 1 2 3 4 1 2 3 4 1 2 3 4 5 6 7 8 1 2 3 4 1 2 3 4 Chapter 1 4 2 0 2 4 6 8 10 12 19. ⏐n 5⏐ 7 {n 2 n 12} 4 3 2 1 0 17. ⏐7r⏐ 14 {r r 2 or r 2} 4 3 2 1 0 15. ⏐t⏐ 3 {t t 3 or t 3} 0 13. 8 3x 2 23 {x 2 x 7} 4 3 2 1 0 11. 10 5x 5 {x 2 x 1} 4 3 2 1 0 or c 0} 9. 2c 1 5 or c 0 {c c 2 8 4 4 8 n 2.5 n 4 1 2 3 4 1 2 3 4 or a 3} 1 2 3 4 numbers 1 1 2 2 3 3 4 4 45 2 4 6 8 Answers 8 6 4 2 0 Glencoe Algebra 2 20. ⏐h 1⏐ 5 {h h 6 or h 4} 4 3 2 1 0 18. ⏐p 2⏐ 2 4 3 2 1 0 16. ⏐6x⏐ 12 {x 2 x 2} 4 3 2 1 0 14. w 4 10 or 2w 6 all real 4 3 2 1 0 12. 4a 8 or a 3 {a a 2 4 3 2 1 0 10. 11 4y 3 1 {y 2 y 1} Solve each inequality. Graph the solution set on a number line. 7. 5. 2 4. all numbers between 6 and 6 n 6 8 6 4 2 0 Write an absolute value inequality for each graph. 4 3 2 1 0 3. all numbers less than 1 or greater than 1 n 1 1 2. all numbers less than 5 and greater than 5 n 5 Answers 1. ⏐3x 4⏐ 8 x 4 x Solve each inequality. Graph the solution set on a number line. Exercises 2 By statement 1 above, if ⏐2x 1⏐ 5, then 5 2x 1 5. Adding 1 to all three parts of the inequality gives 4 2x 6. Dividing by 2 gives 2 x 3. By statement 2 above, if ⏐x 2⏐ 4, then x 2 4 or x 2 4. Subtracting 2 from both sides of each inequality gives x 2 or x 6. 8 6 4 2 0 Solve 2x 1 5. Graph the solution set on a number line. Example 2 Solve x 2 4. Graph the solution set on a number line. 4 3 2 1 0 1. all numbers greater than or equal to 2 or less than or equal to 2 n 2 8:03 AM Example 1 These statements are also true for and . 1. If ⏐a⏐ b, then b a b. 2. If ⏐a⏐ b, then a b or a b. 5/19/06 For all real numbers a and b, b 0, the following statements are true. Use the definition of absolute value to rewrite an absolute value inequality as a compound inequality. Solving Compound and Absolute Value Inequalities Skills Practice Write an absolute value inequality for each of the following. Then graph the solution set on a number line. 1-6 NAME ______________________________________________ DATE______________ PERIOD _____ A2-01-873971 Absolute Value Inequalities 1-6 NAME ______________________________________________ DATE______________ PERIOD _____ Lesson 1-6 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A21 (Lesson 1-6) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Chapter 1 20 10 0 10 20 n 10 4. 4 3 2 1 0 1 2 3 4 8 12 16 20 24 28 32 2 4 6 8 10 12 14 4 3 4 {x x 2 or x 8} 6. 3(5x 2) 24 or 6x 4 4 5x 2 0 3 8 n 2 4 2 3 4 5 5 A22 1 2 3 4 2 4 6 8 10 12 14 16 1 2 1 2 3 3 4 4 1 2 3 4 1 2 3 4 1 2 3 4 1 5 4 3 2 1 0 1 2 3 4 n 16. ⏐3n 2⏐ 2 1 n 3 3 4 3 2 1 0 14. ⏐x⏐ x 1 all real numbers 4 3 2 1 0 1 5 z 12. ⏐2z 2⏐ 3 z 2 2 8 7 6 5 4 3 2 1 0 10. ⏐y 5⏐ 2 {x 7 x 3} 4 3 2 1 0 Chapter 1 46 v 14.5 0.08; {v 14.42 v 14.58} Glencoe Algebra 2 18. MANUFACTURING A company’s guidelines call for each can of soup produced not to vary from its stated volume of 14.5 fluid ounces by more than 0.08 ounces. Write and solve an absolute value inequality to describe acceptable can volumes. r 24 6.5; {r r 17.5 or r 30.5} 17. RAINFALL In 90% of the last 30 years, the rainfall at Shell Beach has varied no more than 6.5 inches from its mean value of 24 inches. Write and solve an absolute value inequality to describe the rainfall in the other 10% of the last 30 years. 4 3 2 1 0 15. ⏐3b 5⏐ 2 4 3 2 1 0 13. ⏐2x 2⏐ 7 5 {x 2 x 0} 0 11. ⏐x 8⏐ 3 {x x 5 or x 11} 4 3 2 1 0 or w 9. ⏐2w⏐ 5 w w 2 2 1 Chapter 1 20N 50,000 100,000 and N 10,000; The attendance must be between 7,500 and 10,000 people, inclusive. 3. CONCERT Jacinta is organizing a large fund-raiser concert in a space with a maximum capacity of 10,000 people. Her goal is to raise at least $100,000. Tickets cost $20 per person. Jacinta spends $50,000 to put the event together. Write and solve a compound inequality that describes N, the number of attendees needed to achieve Jacinta’s goal. n b 2n 2. HIKING For a hiking trip, everybody must bring at least one backpack. However, because of space limitations, nobody is allowed to bring more than two backpacks. Let n be the number of people going on the hiking trip and b be the number of backpacks allowed. Write a compound inequality that describes how b and n are related. 45 d < 55 47 Glencoe Algebra 2 s 12 and 3s 45; s is at least 12 and at most 15 7. Write a compound inequality for s using parts A and B. Find the maximum and minimum values for s. s 12 6. A passenger needs to bring a soil sample on the plane that is at least 1 cubic foot. The passenger is bringing it in a suitcase that is in the shape of a cube with side length s inches. Write an inequality that gives the minimum length for s. h w 45 w 5. Write an inequality that describes the airline’s carry-on size limitation. l h An airline company has a size limitation for carry-on luggage. The limitation states that the sum of the length, width, and height of the suitcase must not exceed 45 inches. AIRLINE BAGGAGE For Exercises 5–7, use the following information. 110 b 110 Answers 4 3 2 1 0 3} 7. 2x 3 15 or 3 7x 17 {x x 2} 8. 15 5x 0 and 5x 6 14 {x x 0 5. 8 3y 20 52 {y 4 y 24} 4 1 2 8:03 AM Solve each inequality. Graph the solution set on a number line. 3. 4 3 2 1 0 8 6 4 2 0 4. NUMBERS Amy is thinking of two numbers a and b. The sum of the two numbers must be within 10 units of zero. If a is between 100 and 100, write a compound inequality that describes the possible values of b. Solving Compound and Absolute Value Inequalities Word Problem Practice 1. AQUARIUM The depth d of an aquarium tank for dolphins satisfies |d 50| 5. Rewrite this as a compound inequality that does not involve the absolute value function. 1-6 ____________ PERIOD _____ 5/19/06 Write an absolute value inequality for each graph. 2. all numbers between 1.5 and 1.5, including 1.5 and 1.5 n 1.5 1. all numbers greater than 4 or less than 4 n 4 6 Solving Compound and Absolute Value Inequalities Practice NAME ______________________________________________ DATE A2-01-873971 Write an absolute value inequality for each of the following. Then graph the solution set on a number line. 1-6 NAME ______________________________________________ DATE______________ PERIOD _____ Lesson 1-6 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A1-A23 Page A22 (Lesson 1-6) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2 Enrichment Chapter 1 Solve 2x 10. x 5 and x 5. x 6 or x 4 3 3x 18 or 3x 4 A23 Chapter 1 x 6 or x 10; disjunction 9. 8 x 2 x 12 or x 16; disjunction 48 Glencoe Algebra 2 x 1 and x 4; conjunction 10. 5 2x 3 x 16 and x 8; conjunction 8. 4 x 3 4 x 7. 1 7 2 6. 7 2x 5 x 1 and x 1; conjunction 1 4 x 2 or x 0; disjunction 4. x 1 1 x 15 and x 1; conjunction 2. x 7 8 x and x ; conjunction 1 2 5. 3x 1 x x 2 and x 3; conjunction 3. 2x 5 1 x 6 or x 6; disjunction 1. 4x 24 Solve each inequality. Then write whether the solution is a conjunction or disjunction. Exercises A compound sentence that combines two statements by the word or is a disjunction. x 6 or x true. Answers Answers 4 3 Every solution for the inequality is a replacement for x that makes either Solve each inequality. Example 2 Solve 3x 7 11. 3x 7 11 means 3x 7 11 or 3x 7 11. A compound sentence that combines two statements by the word and is a conjunction. Every solution for 2x 10 is a replacement for x that makes both x 5 and x 5 true. 8:03 AM Solve each inequality. 2 x 10 means 2x 10 and 2x 10. Example 1 5/19/06 An absolute value inequality may be solved as a compound sentence. A2-01-873971 Conjunctions and Disjunctions 1-6 NAME ______________________________________________ DATE______________ PERIOD _____ A1-A23 Page A23 (Lesson 1-6) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Glencoe Algebra 2