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Name 1-1 Class Date Additional Vocabulary Support Patterns and Expressions Choose the word from the list that best matches each sentence. algebraic expression consecutive diagram pattern variable consecutive 1. Following one another in order. 2. A symbol, usually a letter, that represents one or more numbers. variable 3. A drawing that is included with a math problem. diagram algebraic expression 4. An expression that contains one or more variables. pattern 5. The same type of change between any two shapes or numbers. Use a word from the list above to complete each sentence. 6. The numbers 21 and 22 are consecutive numbers. 7. In the algebraic expression 5x2 2 2x 1 9, x is called the 8. The picture included with a problem is called a variable diagram . . 9. 2x 2 10 is called an algebraic expression . 10. The pattern in a list of numbers is that each one is 5 more than the last one. Describe the pattern for each group of figures. What would the next consecutive figure in the pattern look like? 11. Answers may vary. Sample: a black square in the corner of a larger square moves clockwise from one corner to the next corner; a square with a black square in the lower right corner 12. Answers may vary. Sample: equilateral triangles alternating pointing up, then down; an equilateral triangle pointing up 13. Answers may vary. Sample: alternate a circle inside a square with a square inside a circle; a circle inside a square 14. Answers may vary. Sample: a stack of small squares, ﬁrst one square, then two squares, then 3 squares; a stack of 4 small squares Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 1 Name 1-1 Class Date Think About a Plan Patterns and Expressions Use the graph shown. y 15 a. Identify a pattern of the graph by making a table of the Output inputs and outputs. b. What are the outputs for inputs 6, 7, and 8? 1. What are the ordered pairs of the points in the graph? (1, 3), (2, 6), (3, 9), (4, 12), (5, 15) 12 9 6 3 O x 1 2 3 4 Input 2. Complete the table of the input and output values shown in the ordered pairs. Input Output 1 3 2 6 3 9 4 12 5 15 3. Complete the process column with the process that takes each input value and gives the corresponding output value. 4. output 5 Input Process Column Output 1 1( 3 ) 3 2 2( 3 ) 6 3 3( 3 ) 9 4 4( 3 ) 12 5 5( 3 ) 15 input ∙ 3 5. Complete the process column for inputs 6, 7, and 8. Then find the outputs for inputs 6, 7, and 8. Input Process Column Output 6 6( 3 ) 18 7 7( 3 ) 21 8 8( 3 ) 24 18, 21, 24 6. The outputs for inputs 6, 7, and 8 are Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 2 5 Name Class 1-1 Date Practice Form G Patterns and Expressions Describe each pattern using words. Draw the next figure in each pattern. 1. Answers may vary. Sample: Rotate the previous ﬁgure 90° clockwise. 2. Answers may vary. Sample: Each ﬁgure is obtained by moving the previous corner block up one row and right one column, shading this block and all blocks to the left of and below this new corner block. Answers may vary. Sample: With the center of the middle of the three consecutive blocks as pivot, each ﬁgure is a 90° counterclockwise rotation of the previous ﬁgure. 3. Copy and complete each table. Include a process column. 4. Input Output P. C. 1 4 2 5. Input Output P. C. 5(1) ⫺ 1 1 22 9 5(2) ⫺ 1 2 3 14 5(3) ⫺ 1 4 19 5 6. Input Output P. C. ⫺2(1) 1 0.5 0.5(1) 24 ⫺2(2) 2 1.0 0.5(2) 3 26 ⫺2(3) 3 1.5 0.5(3) 5(4) ⫺ 1 4 28 ⫺2(4) 4 2.0 0.5(4) 24 5(5) ⫺ 1 5 210 ⫺2(5) 5 2.5 0.5(5) 6 29 5(6) ⫺ 1 6 212 ⫺2(6) 6 3.0 0.5(6) n 5n 2 1 5n ⫺ 1 n 22n ⫺2n n 0.5n 0.5n 7. Describe the pattern using words. Answers may vary. Sample: Draw a square. For each subsequent ﬁgure in the pattern, make a new square having vertices at the midpoints of the sides of the previous innermost square, shading all but the new innermost square. Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 3 Name 1-1 Class Date Practice (continued) Form G Patterns and Expressions A gardener plants a flower garden between his house and a brick pathway parallel to the house. The table at the right shows the area of the garden, in square feet, depending on the width of the garden, in feet. Width Area 1 3.5 2 7 3 10.5 4 14 8. What is the area of the garden if the width is 8 feet? 28 ft² 9. What is the area of the garden if the width is 15 feet? 52.5 ft² Identify a pattern and find the next three numbers in the pattern. 10. 25, 210, 220, 240, c Each term is double the previous term; 280, 2160, 2320 11. 5, 8, 11, 14, c Each term is 3 more than the previous term; 17, 20, 23 12. 3, 1, 21, 23, c Each term is 2 less than the previous term; 25, 27, 29 2 3 4 5 14. 3, 4, 5, 6, c The numerator and denominator of each term are each one more than the previous term; 67, 78, 89 13. 1, 3, 6, 10, 15, c Each term is n 1 1 more than the previous term where n is the difference between the previous two terms; 21, 28, 36 15. 10, 9, 6, 1, 26, c Each term is n 1 2 less than the previous term where n is the difference between the previous two terms; 215, 226, 239 The graph shows the cost depending on the number of DVDs that you purchase. $80 $64 Cost 16. What is the cost of purchasing 5 DVDs? $80 17. What is the cost of purchasing 10 DVDs? $160 $48 $32 $16 18. What is the cost of purchasing n DVDs? $16n 0 1 2 3 4 Number of DVDs Keesha earns $320 a week working in a clothing store. As a bonus, her employer pays her $15 more than she earned the previous week, so that at the end of the second week she earns $335, and after 3 weeks, she earns $350. 19. How much will Keesha earn at the end of the fifth week? $380 20. How much will Keesha earn at the end of the tenth week? $455 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 4 5 Name 1-1 Class Date Practice Form K Patterns and Expressions Describe each pattern using words. Draw the next figure in each pattern. 1. Answers may vary. Sample: Start with a row of two squares. Add a row of two squares below the previous ﬁgure for each new ﬁgure. 2. Answers may vary. Sample: Start with one square. Add a row of squares below the previous ﬁgure that has one more square than the row above it for each new ﬁgure. 3. Answers may vary. Sample: Start with an isosceles triangle that points upward. Rotate the triangle 90° clockwise for each new ﬁgure. Make a table with a process column to represent the pattern. Write an expression for the number of circles in the nth figure. The table has been started for you. 4. The expression for the number of circles in the nth ﬁgure is 2n. Figure Number (Input) Process Column Number of Circles (Output) 1 1(2) 2 2 2(2) 4 3 3(2) 6 4 4(2) 8 ■ ■ ■ n n(2) 2n Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 5 Name Class Date Practice (continued) 1-1 Form K Patterns and Expressions Number of cups of flour The graph shows the number of cups of flour needed for baking cookies. 5. How many cups of flour are needed for baking 4 batches of cookies? 8 cups 6. How many cups of flour are needed for baking 30 batches of cookies? 10 Baking Cookies 8 6 4 2 0 0 2 4 6 8 10 Number of batches of cookies 60 cups 7. How many cups of flour are needed for baking n batches of cookies? 2n cups Identify a pattern by making a table. Include a process column. (1, 4), (2, 5), (3, 6), (4, 7) Hint: To start, list the points on the graph. Make a table of input and output values shown in the ordered pairs. Use the process column to figure out the pattern. 10 10 9. 8 8 6 6 Output Output 8. 4 2 4 Input 6 0 8 2 4 Input 6 8 Identify the pattern and find the next three numbers in the pattern. 10. 1, 4, 16, 64, . . . 11. 3, 6, 12, 24, . . . Each term is 4 times the previous term; 256, 1024, 4096 Each term is 2 times the previous term; 48, 96, 192 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 6 Process Column (Output) 1 113 4 2 213 5 3 313 6 4 413 7 n n13 n13 (1, 1.5), (2, 3), (3, 4.5), (4, 6) 2 2 0 4 (Input) (Input) Process Column (Output) 1 1(1.5) 1.5 2 2(1.5) 3 3 3(1.5) 4.5 4 4(1.5) 6 n n(1.5) 1.5n Name Class 1-1 Date Standardized Test Prep Patterns and Expressions Multiple Choice For Exercises 1–5, choose the correct letter. 1. What is the next figure in the pattern at the right? C 2. Which is the next number in the table? H 14 15 16 20 Input Output 1 1 2 3 3 6 4 10 5 ■ 3. How many toothpicks would be in the tenth figure? A 21 20 4. What is the next number in the pattern? 21 23 2, 7, 12, 17, c G 22 5. What is the next number in the pattern? 23 11 23 27 1, 21, 2, 22, 3, c A 0 3 4 Short Response 6. Ramon has 25 books in his library. Each month, he adds 3 new books to his collection. How many books will Ramon have after 12 months? 61 books [2] correct number of books [1] incorrect number of books [0] no answer given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 7 Name Class 1-1 Date Enrichment Patterns and Expressions Often identifying and expressing patterns using algebraic expressions is difficult because the pattern is not immediately obvious. Organizing information into a table and looking at the common differences provide a clue. 1. Copy and complete the table below to find the first differences between consecutive terms. 1 4 Figure Number Number Pattern 2 10 6 6 First Difference 3 16 4 22 5 28 6 6 6 34 6 7 40 6 8 46 6 9 52 6 10 58 6 2. Because the first difference is constant in the pattern above, the algebraic expression will have degree 1. Degree 1 expressions will not have any exponents larger than 1. What is the algebraic expression for the nth term in this pattern? (6n 2 2) 3. Copy and complete the table below by finding the first and second differences between consecutive terms. 1 4 Figure Number Number Pattern 2 10 6 First Difference 8 2 Second Difference 3 18 4 28 10 2 5 40 14 12 2 6 54 2 7 70 16 2 8 88 18 2 20 2 4. Since the second difference is constant in the pattern above, the algebraic expression will have degree 2. In an expression of degree 2, the largest exponent is 2. Therefore, we know that the expression for this pattern must include n2. What is the algebraic expression for the nth term in this pattern? (n2 1 3n) Write an algebraic expression for the nth term in each pattern. 5. 23, 2, 7, 12, 17, c (5n 2 8) 6. 2, 8, 18, 32, 50, c (2n2) 7. 0, 7, 26, 63, 124, c (n3 2 1) 8. 2, 5, 10, 17, 26, c (n2 1 1) Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 8 9 108 10 130 22 2 Name 1-1 Class Date Reteaching Patterns and Expressions Identifying patterns is an important skill in algebra. Identify the underlying structure of a group of numbers, a set of data, or a sequence of figures to express a rule describing the relationship. Problem Look at the figures from left to right. What is the pattern? What would the next figure in the pattern look like? First, identify the basic properties of the figure. The number of elements in each figure increases by one from each figure to the next. Second, determine whether the figures change in size. Notice that the arrows decrease in length by about 13 at each step. Third, observe whether there is displacement or rotation from one figure to the next. Each arrow is a 908 clockwise rotation from the previous arrow. The pattern begins with an arrow pointing to the right. Each subsequent figure adds a new arrow that is shorter than the previous arrow by about 13 and is rotated 908 clockwise from the previous one. The next figure in the pattern is shown at the right. Exercises Describe the pattern in words and draw the next figure in the pattern. 1. Each ﬁgure is twice the size of the one before it. 2. Each ﬁgure adds a new triangle whose dimensions are 50% of the dimensions of the previous smallest triangle and is inside it, rotated 180°. The color alternates white to black from biggest to smallest triangle, respectively. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 9 Name Class 1-1 Reteaching Date (continued) Patterns and Expressions Problem $15 Charge You tell your parents that you will pay the text messaging portion of your cell phone bill. The graph shows the monthly charge depending on the number of text messages you send and receive during the month. How much do you owe if you send and receive a total of 100 text messages during the month? $12 $9 $6 $3 First, identify several points on the graph. Three points on the graph are (10, 3), (20, 6), and (50, 15). Make a table of values for the given inputs. Input Process Column Output 10 10 3 $.30 $3 20 20 3 $.30 $6 50 50 3 $.30 $15 0 10 20 30 40 50 Number of Text Messages Identify the pattern in the process column. Each output is equal to the product of the input and $.30. To summarize the result in words, each text message costs $0.30. Therefore, if you send and receive 100 text messages during the month, then you owe 100 3 $.30 5 $30. Exercises Use the graph at the right to answer the questions. 3. What is the cost to send 10 text messages? 20 text messages? 50 text messages? $5; $10; $25 Charge 4. What is the cost to send one text message? $.50 $25 $20 $15 $10 $5 5. How much do you owe your parents if you send and receive 75 text messages during the month? $37.50 0 10 20 30 40 50 Number of Text Messages Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 10 Name 1-2 Class Date Additional Vocabulary Support Properties of Real Numbers Choose the word from the list that best matches each description. integers irrational numbers natural numbers rational numbers whole numbers whole numbers 1. the natural numbers and zero integers 2. the natural numbers, their opposites, and zero 3. the numbers that can be written as a quotient of integers rational numbers 4. the numbers used for counting natural numbers irrational numbers 5. the numbers that cannot be written as quotients of integers Write all of the numbers from the list that are examples of each subset. 27 6. whole numbers 212 "81 5 7 0 2105 "44 93 27, !81 , 0, 93 7. natural numbers 27, !81 , 93 5 8. rational numbers 27, 212, !81 , 0, 7 , 2105, 93 9. irrational numbers !44 10. integers 27, 212, !81 , 0, 2105, 93 11. Draw a diagram showing the relationship of whole numbers, natural numbers, rational numbers, integers, irrational numbers, and real numbers. Real Numbers Natural numbers Whole numbers Irrational numbers Integers Rational numbers Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 11 Name 1-2 Class Date Think About a Plan Properties of Real Numbers Five friends each ordered a sandwich and a drink at a restaurant. Each sandwich costs the same amount, and each drink costs the same amount. What are two ways to compute the bill? What property of real numbers is illustrated by the two methods? Understanding the Problem 1. There are 5 sandwiches and 5 drinks on the bill. 2. What is the problem asking you to determine? two ways to represent the cost of 5 sandwiches and 5 drinks, and the property of real numbers illustrated by the two representations Planning the Solution 3. How can you represent the cost of five sandwiches? Answers may vary. Sample: 5s 4. How can you represent the cost of five drinks? Answers may vary. Sample: 5d 5. How can you represent the cost of the items ordered by one friend? Answers may vary. Sample: d 1 s Getting an Answer 6. Write an expression that represents the cost of five drinks and the cost of five sandwiches. Answers may vary. Sample: 5d 1 5s 7. Write an expression that represents the cost of the items ordered by five friends. Answers may vary. Sample: 5(d 1 s) 8. What property of real numbers tells you that these two expressions are equal? Explain. Distributive Property; the Distributive Property states a(b 1 c) 5 ab 1 ac. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 12 Name 1-2 Class Date Practice Form G Properties of Real Numbers Classify each variable according to the set of numbers that best describes its values. 1. the area of the circle A found by using the formula pr2 irrational number 2. the number n of equal slices in a pizza; the portion p of the pizza in one slice natural number; rational number 3. the air temperature t in Saint Paul, MN, measured to the nearest degree Fahrenheit integer 4. the last four digits s of a Social Security number natural number 1 ⫺2 2 Graph each number on a number line. ⫺3 ⫺2 ⫺1 5. 21 6. "3 √ 3 2.8 ⫺1 0 7. 2.8 1 2 3 1 8. 22 2 Compare the two numbers. Use + or *. 10. 4, !17 * 9. 2!2, 22 + 11. !29, 5 + 12. !50, 6.8 + 13. 11, !130 * 14. 26, 2!30 * 1 15. 72, !67 * 16. 2!10, 2!12 + Name the property of real numbers illustrated by each equation. 17. 2(3 1 !5) 5 2 ? 3 1 2 ? !5 18. 16 1 (213) 5 213 1 16 Distributive Property 1 19. 27 ? 27 5 1 Inverse Property of Multiplication Commutative Property of Addition 20. 5(0.2 ? 7) 5 (5 ? 0.2) ? 7 Associative Property of Multiplication Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 13 Name 1-2 Class Date Practice (continued) Form G Properties of Real Numbers Estimate the numbers graphed at the labeled points. A B ⫺3 ⫺2 ⫺1 C 0 1 D 2 3 21. point A Answers may vary. Sample: 22.7 22. point B Answers may vary. Sample: 21.2 23. point C Answers may vary. Sample: 0.9 24. point D Answers may vary. Sample: 3.0 Geometry To find the length of side b of a rectangular prism with a square base, use the formula b 5 ÅVh , where V is the volume of the prism and h is the height. Which set of numbers best describes the value of b for the given values of V and h? 25. V 5 100, h 5 5 irrational numbers 26. V 5 100, h 5 25 natural numbers 27. V 5 100, h 5 20 irrational numbers 28. V 5 5, h 5 20 rational numbers h Write the numbers in increasing order. 4 5 29. 2"2, 5, 24, 0.9, 21 2 54, 21, 45, 0.9, 2"2 5 2 30. 8, 26, 3, 2p, 20.5 26, 2π, 20.5, 58, 23 Justify the equation by stating one of the properties of real numbers. 31. (x 1 37) 1 (237) 5 x 1 (37 1 (237)) Associative Property of Addition 32. x ? 1 5 x Identity Property of Multiplication 33. x 1 (37 1 (237)) 5 x 1 0 Inverse Property of Addition 34. x 1 0 5 x Identity Property of Addition Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 14 b b Name Class 1-2 Date Practice Form K Properties of Real Numbers Classify each variable according to the set of numbers that best describes its values. 1. the number of students in your class natural numbers To start, make a list of some numbers that could describe the number of students in your class. 2. the area of the circle A found by using the formula A 5 pr2 irrational numbers To start, make a list of some numbers that could describe the area of a circle. 3. the elevation e of various land points in the United States measured to the nearest foot integers To start, make a list of some numbers that could describe elevation levels. Graph each number on a number line. 1 4. 5 2 6. 2.25 5 ⫺4 5. 24 1 2 ⫺8⫺7⫺6⫺5⫺4⫺3⫺2⫺1 0 1 2 ⫺1 0 1 2 3 4 5 6 7 8 9 2.25 1 7. 26 3 ⫺1 0 1 2 3 4 5 6 7 8 9 ⫺6 1 3 ⫺8⫺7⫺6⫺5⫺4⫺3⫺2⫺1 0 1 2 8. !8 √ 8 ⬇ 2.83 To start, use a calculator to approximate the square root. ⴚ1 0 1 2 3 4 5 6 7 8 9 Compare the two numbers. Use R or S . 9. !50 and 8.8 R 10. 5 and !23 S 11. 6.2 and !40 R 12. 2!3 and 23 S Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 15 Name 1-2 Class Date Practice (continued) Form K Properties of Real Numbers Name the property of real numbers illustrated by each equation. 2 3 13. 3 ? 2 5 1 Inverse Property of Multiplication 14. 6(2 1 x) 5 6 ? 2 1 6 ? x Distributive Property 15. 2 ? 20 5 20 ? 2 Commutative Property of Multiplication 16. 8 1 (28) 5 0 Inverse Property of Addition 17. 2(0.5 ? 4) 5 (2 ? 0.5) ? 4 Associative Property of Multiplication 18. 211 1 5 5 5 1 (211) Commutative Property of Addition Estimate the numbers graphed at the labeled points. C A ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 19. point A Answers may vary. Sample: 22 B 0 1 D 2 3 4 5 20. point B Answers may vary. Sample: 1.5 21. point C Answers may vary. Sample: 24.75 22. point D Answers may vary. Sample: 2.25 To find the length of the side b of the square base of a rectangular prism, use the formula b 5 ÅVh , where V is the volume of the prism and h is the height. Which set of numbers best describes the value of b for the given values of V and h? 23. V 5 100, h 5 1 24. V 5 100, h 5 10 natural numbers irrational numbers Write the numbers in increasing order. 5 5 25. 6, !28, 22, 20.8, 1 2 26. 3, 24, !32, !13, 20.4 24, 20.4, 23, !13, !32 252, 20.8, 56, 1, "28 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 16 Name Class 1-2 Date Standardized Test Prep Properties of Real Numbers Multiple Choice For Exercises 1–5, choose the correct letter. 1. Which letter on the graph corresponds to !5? C 2. Which letter on the graph corresponds to 21.5? F A ⫺1 BC 0 1 F 2 G ⫺3 ⫺2 ⫺1 D 3 H 0 4 5 2 3 I 1 What property of real numbers is illustrated by the equation? 3. 26 1 (6 1 5) 5 (26 1 6) 1 5 D Identity Property of Addition Commutative Property of Addition Inverse Property of Addition Associative Property of Addition 4. 2(24 1 x) 5 2(24) 1 2 ? x G Associative Property of Multiplication Associative Property of Addition Distributive Property Closure Property of Multiplication 2 5. Which of the following shows the numbers 13, 1.3, 17, 24, and 2!10 in order from greatest to least? B 13, 1.3, 127, 24, 2!10 13, 127, 1.3, 2!10, 24 13, 1.3, 127, 2!10, 24 24, 2!10, 127, 1.3, 13 Short Response Geometry The length c of the hypotenuse of a right triangle with legs having lengths a and b is found by using the formula c 5 "a2 1 b2 . Which set of numbers best describes the value of c for the given values of a and b? 6. a 5 3, b 5 4 [2] natural numbers [1] incorrect set of numbers [0] no answer given 1 1 7. a 5 3, b 5 4 [2] rational numbers [1] incorrect set of numbers [0] no answer given 8. a 5 !3, b 5 !4 [2] irrational numbers [1] incorrect set of numbers [0] no answer given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 17 Name 1-2 Class Date Enrichment Properties of Real Numbers There are four words beginning with the letter “I” that describe certain types of operations: a. Identity: an operation that does not change anything. For example, adding 0 to a number is an identity operation, because adding 0 does not change the original number. b. Inverse: an operation that can be undone by another operation. For instance, the operation of adding 2 to a number can be undone by subtracting 2 from the number. However, the operation of multiplying a number by 0 cannot be undone. c. Idempotent: an operation that, when done twice, is the same as doing it once. For example, multiplying a number by 1 and then multiplying the result by 1 again has exactly the same effect as multiplying the number by 1 only once. d. Involutory: an operation that, when done twice, leaves a number unchanged. For instance, multiplying a number by 21 and then multiplying the result by 21 again returns the original number. For each of the following operations, state which of the “I” words apply. If none apply, write none. 1. finding the absolute value of a number idempotent 2. dividing a number by 1 identity, inverse, idempotent, involutory 3. multiplying the absolute value of a number by 21 idempotent 4. finding the reciprocal of a nonzero number inverse, involutory 5. dividing a number by 21 inverse, involutory 6. multiplying a number by 0 idempotent 7. adding 0 to the reciprocal of a nonzero number inverse, involutory 8. multiplying the reciprocal of a nonzero number by 2 inverse, involutory 9. adding the absolute value of a nonzero number to the absolute value of its reciprocal none 10. finding the reciprocal of the absolute value of the reciprocal of a nonzero number idempotent 11. finding the absolute value of 21 times a nonzero number, then taking the reciprocal none Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 18 Name Class 1-2 Date Reteaching Properties of Real Numbers The Properties of Real Numbers are relationships that are true for all real numbers except zero. The additive identity for real numbers is 0. This gives the Identity Property of Addition, which states for any real number a: a 1 0 5 a and 0 1 a 5 a The additive inverse of a real number a is 2a. By the Inverse Property of Addition: a 1 (2a) 5 0 There are two similar properties for multiplication. These use the multiplicative identity 1 and the multiplicative inverse a1 for any nonzero real number a. Identity Property of Multiplication: a ? 1 5 a and 1 ? a 5 a 1 Inverse Property of Multiplication: a ? a 5 1 Problem Using the Properties of Real Numbers, what is the missing number in the equation? a. u1055 According to the Identity Property of Addition, the missing number is 5. b. 7 ? u51 The Inverse Property of Multiplication shows that the product of a real number and its multiplicative inverse is 1. The missing number is the multiplicative inverse of 7, or 17 . Exercises Find the missing number in the equation. 1. 0 1 (24) 5 24 u 2. 22 1 u 3. 1 ? 22 5 22 4. 2 50 u u ? 32 5 1 2 3 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 19 Name Class 1-2 Reteaching Date (continued) Properties of Real Numbers The Commutative and Associative Properties of Addition and Multiplication are properties that help you simplify calculations. The Commutative Property states that the order of addition or multiplication does not change the sum or product. a1b5b1a ab 5 ba The Associative Property states that the grouping of three or more addends or factors does not change the sum or product. (a 1 b) 1 c 5 a 1 (b 1 c) (ab)c 5 a(bc) Problem What property does the equation illustrate? 1 1 5 ? Q 5 ? 85 R 5 Q 5 ? 5 R ? 85 This equation shows that the product of three numbers is the same regardless of the order of multiplication. Only the grouping of the factors is different. Therefore, the equation illustrates the Associative Property of Multiplication. Exercises Name the property that the equation illustrates. 1 5. Q 5 ? 5 R ? 85 5 1 ? 85 Inverse Property of Multiplication 6. 1 ? 85 5 85 Identity Property of Multiplication The Distributive Property combines addition and multiplication: a(b 1 c) 5 ab 1 ac. Problem What are the missing values in the equation? 4 ? (6 1 3) 5 4 ? u14?u By the Distributive Property, the sum of two numbers multiplied by a third number is equal to the sum of each multiplied by the third number. Because the third number is 4, the missing numbers are 6 and 3. Exercises Name the missing values in each equation. 3 ?2 u1u 2 5u 9 ( y 1 2) 9. 9 ? y 1 9 ? u 22 ? 5 2 2 ? 1 u 1 1b 5 u 8. 22 a 5 7. 3(a 1 2) 5 3 ? a z z 1x b 10. 8(23) 1 8 ? x 5 8 a 23 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 20 Name 1-3 Class Date Additional Vocabulary Support Algebraic Expressions Use the chart below to review vocabulary. These vocabulary words will help you complete this page. Addition (1) Subtraction (2) Multiplication (3) Division (4) sum difference product quotient more than less than times divided by increased by fewer than total subtracted from added to Circle the word or words in each word phrase that tell you what operations to use. Write the operation symbol word (1, 2, 3, 4) next to the algebraic expression. 1. the sum of a number m and 212 1 2. the product of b and c 3 3. 14 less than p 2 4. the total of 275 and t 1 5. the quotient of d and 28 4 Match each word phrase in Column A with the matching algebraic expression in Column B. Column A Column B 6. the difference of a number p and 36 A. y 1 9 7. 15 more than the number q B. 10(r) 8. the product of 10 and a number r C. q 1 15 9. the total of a number y and 9 D. p 2 36 Match each algebraic expression in Column A with the matching word phrase in Column B. Column A 10. m 1 45 C m Column B A. 45 less than a number m 11. 45 D B. 45 times the sum of a number m and 1 12. m 2 45 A C. a number m increased by 45 13. 45(m 1 1) B D. a number m divided by 45 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 21 Name 1-3 Class Date Think About a Plan Algebraic Expressions Write an algebraic expression to model the situation. The freshman class will be selling carnations as a class project. What is the class’s income after it pays the florist a flat fee of $200 and sells x carnations for $2 each? 1. What does the variable represent? the number of carnations sold 2. How will the class’s income change for each carnation sold? It will increase by $2. 3. Will paying the florist increase or decrease their income? By how much? decrease; $200 4. Will the expression include both the income for each carnation and the florist’s fee? Explain. Yes; the class’s proﬁt is a function of the proceeds from carnation sales and the ﬂorist’s fee. 5. Write the expression in words. The income z 2200 z is and 2 z z times x z . z 6. Write the expression using symbols. income 5 z 2200 z 1 z z 2 z z . z z x z z 7. Check your expression by substituting 300 for the number of carnations. Does your answer make sense? Explain. 2200 1 2(300) 5 2200 1 600 5 400; yes; the class has an income of $400 after they make $600 and pay the ﬂorist $200. z z 220012x 8. The algebraic expression models the freshman class income. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 22 Name 1-3 Class Date Practice Form G Algebraic Expressions Write an algebraic expression that models each word phrase. 1. seven less than the number t t 2 7 2. the sum of 11 and the product of 2 and a number r 11 1 2r Write an algebraic expression that models each situation. 3. Arin has $520 and is earning $75 each week babysitting. 520 1 75w 4. You have 50 boxes of raisins and are eating 12 boxes each month. 50 2 12m Evaluate each expression for the given values of the variables. 5. 24v 1 3(w 1 2v) 2 5w; v 5 22 and w 5 4 212 6. c(3 2 a) 2 c2 ; a 5 4 and c 5 21 0 7. 2(3e 2 5f ) 1 3(e2 1 4f ); e 5 3 and f 5 25 35 Surface Area The expression 6s2 represents the surface area of a cube with edges of length s. What is the surface area of a cube with each edge length? 8. 3 inches 54 in.2 9. 1.5 meters 13.5 m2 The expression 4.95 1 0.07x models a household’s monthly long-distance charges, where x represents the number of minutes of long-distance calls during the month. What are the monthly charges for each number of long-distance minutes? 10. 73 minutes $10.06 11. 29 minutes $6.98 Simplify by combining like terms. 3(a 2 b) 1 49 b 13a 1 19b 9 12. 5x 2 3x2 1 16x2 13x2 1 5x 13. t2 14. t 1 2 1 t2 1 t 32t2 1 2t 15. 4a 2 5(a 1 1) 2a 2 5 16. 22( j2 2 k) 2 6( j2 1 3k) 28j2 2 16k 17. x(x 2 y) 1 y(y 2 x) x2 2 2xy 1 y2 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 23 Name 1-3 Class Practice Date Form G (continued) Algebraic Expressions 18. In a soccer tournament, teams receive 6 points for winning a game, 3 points for tying a game, and 1 point for each goal they score. What algebraic expression models the total number of points that a soccer team receives in a tournament? Suppose one team wins two games and ties one game, scoring a total of five goals. How many points does the team receive? 6w 1 3t 1 1g; 20 points Evaluate each expression for the given value of the variable. 19. 2t2 2 (3t 1 2); t 5 5 242 20. i2 2 5(i3 2 i2); i 5 4 2224 a⫺b 21. Perimeter Write an expression for the perimeter of the figure c at the right as the sum of the lengths of its sides. What is the simplified form of this expression? a 1 (a 2 b) 1 c 1 b 1 (a 2 2c) 1 b 1 c 1 (a 2 b); 4a b a ⫺ 2c a b 22. Simplify 2(2x 2 5y) 1 3(4x 1 2y) and justify each step in your simplification. 22x 1 5y 1 12x 1 6y, Opposite of a c Difference and Distributive Property; 10x 1 11y; Combine like terms a⫺b using the Distributive Property 23. Error Analysis Alana simplified the expression as shown. Do you agree 2(x x + 4) - (5x x - 7) with her work? Explain. No; Dist. Prop. and Opp. of a Diff. x + 4 - 5x x-7 2x incorrectly applied. Correct simpliﬁcation: 2x 1 8 2 5x 1 7 5 23x 1 15 -3x x-3 24. Open-Ended Write an example of an algebraic expression that always has the same value regardless of the value of the variable. Answers may vary. Sample: any expression that results in the variable having a coefﬁcient of 0; sample: x 2 x Match the property name with the appropriate equation. 25. Opposite of a Difference E A. 2f(2r) 1 2pg 5 2(2r) 2 2p 26. Opposite of a Sum A B. 16d 2 (3d 1 2)(0) 5 16d 2 0 27. Opposite of an Opposite F C. 5(2 2 x) 5 10 2 5x 28. Multiplication by 0 B D. 2(4r 1 3s) 1 t 5 (21)(4r 1 3s) 1 t 29. Multiplication by 21 D E. 2(8 2 3m) 5 3m 2 8 30. Distributive Property C F. 2f2(9 2 2w)g 5 9 2 2w Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 24 Name 1-3 Class Date Practice Form K Algebraic Expressions Write an algebraic expression that models each word phrase. 1. six less than the number r r 2 6 To start, relate what you know. “Less than” means subtraction. Describe what you need to find. Begin with the number r and subtract 6. 2. twelve more than the number b b 1 12 3. five times the sum of 3 and the number m 5(3 1 m) Write an algebraic expression that models each situation. 4. Alexis has $250 in her savings account and deposits $20 each week for w weeks. 250 1 20w 5. You have 30 gallons of gas and you use 5 gallons per day for d days. 30 2 5d Evaluate each expression for the given values of the variables. 6. 22a 1 5b 1 6a 2 2b 1 a; a 5 23 and b 5 2 29 To start, substitute the value for each variable. 22(23) 1 5(2) 1 6(23) 2 2(2) 1 (23) 7. y(3 2 x) 1 x2; x 5 2 and y 5 12 16 8. 3(4e 2 2f ) 1 2(e 1 8f ); e 5 23 and f 5 10 58 The expression 6s2 represents the surface area of a cube with edges of length s. What is the surface area of a cube with each edge length? 9. 4 centimeters 96 cm2 10. 2.5 feet 37.5 ft2 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 25 Name 1-3 Class Date Practice (continued) Form K Algebraic Expressions Write an algebraic expression to model the total score in each situation. Then evaluate the expression to find the total score. 11. In the first half, there were fifteen two-point shots, ten three-point shots and 5 one-point free throws. T 5 2w 1 3r 1 1f; 65 To start, define your variables. Let w 5 the number of two-point shots, r 5 the number of three-point shots, and f 5 the number of one-point free throws. 12. In the first quarter, there were two touchdowns and 1 extra point kick. T 5 6t 1 1e; 13 Hint: A touchdown is worth 6 points. An extra point kick is worth 1 point. Simplify by combining like terms. 13. 10b 2 b 9b 14. 12 1 8s 2 3s 12 1 5s 15. 3a 1 2b 1 6a 9a 1 2b 16. 5m 1 2n 1 6m 1 4n 11m 1 6n 17. 8r 2 (3s 2 5r) 13r 2 3s 18. 2.5y 2 4y 21.5y The expression 19.95 1 0.05x models a household’s monthly Internet charges, where x represents the number of online minutes during the month. What are the monthly charges for each number of online minutes? 19. 65 minutes $23.20 20. 128 minutes $26.35 Evaluate each expression for the given value of the variable. 21. 3a 1 (2a 1 6); a 5 2 16 22. x 2 5(x 1 2); x 5 25 10 23. 2r 1 (3r2 1 1); r 5 4 45 24. x2 2 5(3x 2 12); x 5 10 10 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 26 Name Class 1-3 Date Standardized Test Prep Algebraic Expressions Multiple Choice For Exercises 1–3, choose the correct letter. 1. The expression 2p(rh 1 r2) represents the total surface area of a cylinder with height h and radius r. What is the surface area of a cylinder with height 6 centimeters and radius 2 centimeters? C 16p cm2 32p cm2 28p cm2 96p cm2 2. Which expression best represents the simplified form of 3(m 2 3) 1 m(5 2 m) 2 m2 ? F 22m2 1 8m 2 9 22m2 2 2m 2 9 8m 2 9 22m 2 9 3. The price of a discount airline ticket starts at $150 and increases by $30 each week. Which algebraic expression models this situation? D 30 1 150w 30 2 150w 150 2 30w 150 1 30w Extended Response 4. Members of a club are selling calendars as a fundraiser. The club pays $100 for a box of wall and desk calendars. They sell wall calendars for $12 and desk calendars for $8. a. Write an algebraic expression to model the club’s profit from selling w wall calendars and d desk calendars. Explain in words or show work for how you determined the expression b. What is the club’s profit from selling 9 wall calendars and 7 desk calendars? Show your work. a. The income from wall calendars is 12w. The income from desk calendars is 8d. The total income is 12w 1 8d. The club must pay $100 from the total income, so the proﬁt is 12w 1 8d 2 100. (OR equivalent explanation) b. 12w 1 8d 2 100 5 12(9) 1 8(7) 2 100 5 108 1 56 2 100 5 64; $64 [4] appropriate methods and correct expression with no computational errors [3] appropriate methods and correct expression, but minor computational error [2] incorrect expression or multiple computational errors [1] correct expression and proﬁt, without work shown [0] no answer or no attempt made Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 27 Name 1-3 Class Date Enrichment Algebraic Expressions Math Puzzle Algebraic expressions can help reveal the secret behind number puzzles that appear to be magic. Start by trying the puzzle below. Complete the puzzle two times and record each step in the table. First Guess Directions Second Guess Think of a number. Add 1 to your number. Multiply your answer by 2. Subtract 2 from your answer. Finally, subtract your original number. What did you notice about your final answer each time? Answers may vary. Sample: The ﬁnal number is the same as the original number. By writing and simplifying algebraic expressions, you can explain the puzzle. Let n represent your number. Write an algebraic expression for each of the steps in the puzzle above. 1. Add 1 to your number. n 1 1 2. Multiply your answer by 2. 2n 1 2 3. Subtract 2 from your answer. 2n 4. Subtract your original number. n 5. Explain how your algebraic expressions show why the puzzle always ends with the original number. Answers may vary. Sample: The ﬁnal expression is the same as the original expression. Write and simplify algebraic expressions for the puzzle below. 6. Think of a number. n 7. Multiply your number by 4. 4n 8. Subtract 2. 4n 2 2 9. Divide your number by 2. 2n 2 1 10. Add 1. 2n 11. Subtract your original number. n Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 28 Name Class Date Reteaching 1-3 Algebraic Expressions You can model words with algebraic expressions. In a word problem, look for words and word phrases that indicate mathematical operations. Addition added to plus sum more than longer than increased by total in all Subtraction subtracted from minus difference less than shorter than decreased by fewer than Multiplication multiplied by product times of Division divided by quotient fraction of per Problem What is an algebraic expression that models the given word phrase? The quotient of 4 more than the number z and the number y decreased by 3 division 4 addition z 4 (4 1 z) (41z) y subtraction 3 (y 4 2 3) (y23) (41z) (y23) Exercises Write an algebraic expression that models each word phrase. 1. nine less than 5 multiplied by the number p 5p 2 9 2. the product of 2 divided by the number h and 8 more than the number k Q h2 R (k 1 8) 3. two decreased by the quotient of the number a and 7 and increased by a multiplied by 3 2 2 a7 1 3a Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 29 Name Class 1-3 Reteaching Date (continued) Algebraic Expressions To simplify an algebraic expression, combine like terms using the basic properties of real numbers. Like terms have the same variables raised to the same powers. To evaluate an algebraic expression, replace the variables in the expression with numbers and follow the order of operations. Problem What is the value of the algebraic expression 3(4x 1 5y) 2 2(3x 2 7y) when x 5 3 and y 5 22? Simplify the algebraic expression using the basic properties of real numbers. 3(4x 1 5y) 2 2(3x 2 7y) 5 12x 1 15y 2 2(3x 2 7y) Distributive Property for Addition 5 12x 1 15y 2 (6x 2 14y) Distributive Property for Subtraction 5 12x 1 15y 2 6x 1 14y Opposite of a Difference 5 12x 2 6x 1 15y 1 14y Identify like terms. 5 (12 2 6)x 1 (15 1 14)y Distributive Property 5 6x 1 29y Combine like terms. Evaluate the expression, replacing x with 3 and y with 22 in the simplified expression. 6(3) 1 29(22) 5 18 2 58 5 240 Exercises Simplify the algebraic expression. Then evaluate the simplified expression for the given values of the variable. 4. (4x 1 1) 1 2x; x 5 3 5. 7(t 1 3) 2 11; t 5 4 6x 1 1; 19 7t 1 10; 38 6. 3y 1 4z 1 6y 2 9z; y 5 2, z 5 1 7. 2(u 1 v) 2 (u 2 v); u 5 8, v 5 23 9y 2 5z; 13 8. 5a2 u 1 3v; 21 9. 6p2 2 (3p2 1 2q2); p 5 1, q 5 5 3p2 2 2q2; 247 s 1 r r 11. 2 1 3 2 4 1 5; r 5 21, s 5 0 1 5a 1 a 1 1; a 5 22 5a2 1 6a 1 1; 9 3 1 10. 4(m 1 n) 2 4(m 2 n); m 5 6, n 5 2 1 2m r 4 1 n; 5 1 1 3s 1 15; 220 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 30 Name Class 1-4 Date Additional Vocabulary Support Solving Equations The column on the left shows the steps used to solve a problem with an equation. Use the column on the left to answer each question in the column on the right. Problem 1. Read the title of the Problem. What Solve by Setting up and Solving an Equation Two planes leave San Antonio at the same time. The northbound plane travels 70 mi/h faster than the southbound plane. The planes are 1940 mi apart in 2 h. How fast is the southbound plane flying? Relate process are you going to use to solve the problem? Answers may vary. Sample: setting up and solving an equation 2. What is the formula that relates total distance the distance the distance southbound northbound 1 5 between the plane travels plane travels planes after in 2 h in 2 h 2h Deﬁne distance, rate, and time? d 5 rt 3. Why can you represent the rate Let x 5 the rate of the southbound plane. Let x 1 70 5 the rate of the northbound plane. of the northbound plane with the algebraic expression x 1 70? It travels 70 mi/h faster than the southbound plane. Write 4. What does the expression 2x represent? 2x 1 2(x 1 70) 5 1940 2x 1 2x 1 140 5 1940 4x 1 140 5 1940 4x 1 140 2 140 5 1940 2 140 4x 5 1800 It represents the distance the Dist. Property Combine like terms. Subtract. Simplify. Calculate southbound plane travels in 2 h. 5. Why do you divide both sides by 4? to isolate x 6. What does x represent? 4x 1800 4 5 4 the speed of the southbound plane Divide. x 5 450 Answer the question asked. 7. How can you find the speed of the northbound plane? The southbound plane is flying at 450 mi/h. Add 70 mi/h to the speed of the southbound plane. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 31 Name 1-4 Class Date Think About a Plan Solving Equations Geometry The measure of the supplement of an angle is 208 more than three times the measure of the original angle. Find the measures of the angles. Know z z 1808 . 1. The sum of the measures of the two angles is 2. What do you know about the supplemental angle? It is 208 more than three times the measure of the original angle. Need 3. To solve the problem, I need to define: the measure of the original angle 5 x the measure of the supplemental angle 5 3x 1 20 Plan 4. What equation can you use to find the measure of the original angle? x 1 (3x 1 20) 5 180 5. Solve the equation. x 5 40 6. What are the measures of the angles? original angle: 408; supplemental angle: 1408 7. Are the solutions reasonable? Explain. Yes; three times the measure of the ﬁrst angle plus 208 is 1408; the sum of the measures of the angles is 1808, so they are supplementary. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 32 Name 1-4 Class Date Practice Form G Solving Equations Solve each equation. 1. 7.2 1 c 5 19 11.8 2. 8.5 5 5p 1.7 d 3. 4 5 231 2124 4. s 2 31 5 20.6 51.6 Solve each equation. Check your answer. 5. 9(z 2 3) 5 12z 29 6. 7y 1 5 5 6y 1 11 6 7. 5w 1 8 2 12w 5 16 2 15w 1 8. 3(x 1 1) 5 2(x 1 11) 19 Write an equation to solve each problem. 9. Two brothers are saving money to buy tickets to a concert. Their combined savings is $55. One brother has $15 more than the other. How much has each saved? Variable may vary. Sample: s 1 s 1 15 5 55 10. Geometry The sides of a triangle are in the ratio 5 : 12 : 13. What is the length of each side of the triangle if the perimeter of the triangle is 15 inches? Variable may vary. Sample: 5x 1 12x 1 13x 5 15 11. What three consecutive numbers have a sum of 126? Variable may vary. Sample: n 1 (n 1 1) 1 (n 1 2) 5 126 Determine whether the equation is always, sometimes, or never true. 12. 6(x 1 1) 5 2(5 1 3x) never 13. 3(y 1 3) 1 5y 5 4(2y 1 1) 1 5 always Solve each formula for the indicated variable. 14. S 5 L(1 2 r), for r r 5 1 2 SL 2 lh 15. A 5 lw 1 wh 1 lh, for w w 5 Al 1 h Solve each equation for y. 4 16. 9 (y 1 3) 5 g y 5 94 g 2 3 18. c (a 1 b) 17. a(y 1 c) 5 b(y 2 c) y 5 b 2 a , a u b y13 2 3 t 5 t y 5 t 2 3, t u 0 19. 3y 2 yz 5 2z y 5 3 2z 2 z, z u 3 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 33 Name 1-4 Class Date Practice (continued) Form G Solving Equations Solve each equation. 20. 0.5(x 2 3) 1 (1.5 2 x) 5 5x 0 21. 1.2(x 1 5) 5 1.6(2x 1 5) 21 22. 0.5(c 1 2.8) 2 c 5 0.6c 1 0.3 1 u u u 23. 5 1 10 2 6 5 1 15 2 Solve each formula for the indicated variable. p 24. V 5 3 r2h, for h h 5 3V2 πr 25. D 5 kA c T2 2 T1 DL d for T1 T1 5 T2 2 kA L Write an equation to solve each problem. 26. Two trains left a station at the same time. One traveled north at a certain speed and the other traveled south at twice that speed. After 4 hours, the trains were 600 miles apart. How fast was each train traveling? Variable may vary. Sample: 4r 1 4(2r) 5 600 27. Geometry The sides of one cube are twice as long as the sides of a second cube. What is the side length of each cube if the total volume of the cubes is 72 cm3? Variable may vary. Sample: s3 1 (2s)3 5 72 28. Error Analysis Brenna solved an equation for m. Do you agree with her? Explain your answer. No; there is an m on both sides of the equation; v2 mv1 = (m M)v mv m +M mv2 + M Mv v2 m= v1 Mv the correct result should be m 5 v1 2 2v2 Solve each problem. 29. You and your friend left a bus terminal at the same time and traveled in opposite directions. Your bus was in heavy traffic and had to travel 20 miles per hour slower than your friend’s bus. After 3 hours, the buses were 270 miles apart. How fast was each bus going? Your bus: 35 mi/h; Your friend’s bus: 55 mi/h 30. Geometry The length of a rectangle is 5 centimeters greater than its width. The perimeter is 58 centimeters. What are the dimensions of the rectangle? w 5 12 cm, l 5 17 cm 31. What four consecutive odd integers have a sum of 336? 81, 83, 85, 87 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 34 Name 1-4 Class Date Practice Form K Solving Equations Solve each equation. 1. 5x 1 4 5 2x 1 10 2 2. 10w 2 3 5 8w 1 5 4 To start, subtract 2x from each side. To start, subtract 8w from each side. 3. 4(d 2 3) 5 2d 6 4. s 1 2 2 3s 2 16 5 0 27 Solve each equation. Check your answer. 5. 9(z 2 3) 5 12z 29 6. 7y 1 5 5 6y 1 11 6 7. 5w 1 8 2 12w 5 16 2 15w 1 8. 3(x 1 1) 5 2(x 1 11) 19 Write an equation to solve each problem. 9. Lisa and Beth have babysitting jobs. Lisa earns $30 per week and Beth earns $25 per week. How many weeks will it take for them to earn a total of $275? To start, record what you know. Describe what you need to find. Lisa earns $30 per week. Beth earns $25 per week. Total earned: $275 an equation to find the number of weeks it takes to earn $275 together Variable may vary. Sample: 30w 1 25w 5 275 10. The angles of a triangle are in the ratio 2 : 12 : 16. The sum of all the angles in a triangle must equal 180 degrees. What is the degree measure of each angle of the triangle? Let x 5 the common factor. 2x 1 12x 1 16x 5 180 11. What two consecutive numbers have a sum of 53? Variable may vary. Sample: n 1 (n 1 1) 5 53 Determine whether the equation is always, sometimes, or never true. 12. 3(2x 2 4) 5 6(x 2 2) always 13. 4(x 1 3) 5 2(2x 1 1) never Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 35 Name Class Date Practice (continued) 1-4 Form K Solving Equations Solve each formula for the indicated variable. 1 14. A 5 2bh, for b b 5 2A h 15. P 5 2w 1 2l, for w 1 16. A 5 2h(b1 1 b2), for h h 5 b 2A 1 b 17. S 5 2prh 1 2pr2 , for h 1 w 5 P 22 2l 2pr2 h 5 S 22pr 2 Solve each equation for y. 18. ry 2 sy 5 t y 5 r 2t s, r u s 3 19. 7(y 1 2) 5 g y 5 73 g 2 2 y 20. m 1 3 5 n y 5 m(n 2 3), m u 0 21. 3y 2 1 5z 2 2z 1 1 y5 3 Solve each equation. 22. (x 2 3) 2 2 5 6 2 2(x 1 1) 3 23. 4(a 1 2) 2 2a 5 10 1 3(a 2 3) 7 24. 2(2c 1 1) 2 c 5 213 25 25. 8u 1 2(u 2 10) 5 0 2 26. The first half of a play is 35 minutes longer than the second half of the play. If the entire play is 155 minutes long, how long is the first half of the play? Write an equation to solve the problem. Variable may vary. Sample: (m 1 35) 1 m 5 155; m 5 60 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 36 Name Class Date Standardized Test Prep 1-4 Solving Equations Gridded Response Solve each exercise and enter your answer in the grid provided. 1. A bookstore owner estimates that her weekly profits p can be described by the equation p 5 8b 2 560, where b is the number of books sold that week. Last week the store’s profit was $720. What is the number of books sold? 2. What is the value of m in the equation 0.6m 2 0.2 5 3.7? 3. Three consecutive even integers have a sum of 168. What is the value of the largest integer? 4. If 6(x 2 3) 2 2(x 2 2) 5 11, what is the value of x? 5. Your long distance service provider charges you $.06 per minute plus a monthly access fee of $4.95. For referring a friend, you receive a $10 service credit this month. If your long-distance bill is $7.85, how many long-distance minutes did you use? Answers 1. – 160 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 2. 0 1 2 3 4 5 6 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 – 6 . 5 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 3. 0 1 2 3 4 5 6 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 – 58 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 4. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 – 6 . 25 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 5. 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 37 – 215 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 Name 1-4 Class Date Enrichment Solving Equations Equations can be subdivided into three distinct types: a. conditional equations, or equations that are true for some values of x. For example, the equation x 1 1 5 0 is true only for x 5 21. b. identities, for which every possible value of the variable belongs to the solution set. For example, the equation x 5 x is an identity, as it is true for all values of x. c. impossibilities, for which no possible values of the variable belong to the solution set. For example, the equation x 5 x 1 1 is an impossibility, as it is never true. For each of the following equations, find the solution if it is a conditional equation, or classify the equation as an identity or an impossibility. 1. x 1 (2x 2 4) 5 11 5 2. x 5 x 1 2 impossibility 3. x 1 (2x 2 1) 5 3x 2 1 identity 4. x 5 (2 1 2x) 2 x impossibility 5. (x 2 2) 1 (2x 1 4) 5 x 21 6. x 1 2 5 x 1 3 impossibility 7. 2x 5 3x 0 2 8. 2x 1 8 5 6 2 x 23 1 9. 2x 2 4 1 3x 5 8x 2 5 3 10. 2x 1 5 5 5 1 2x identity 11. 2(x 1 3) 5 5x 2 (3x 2 6) identity 12. x 1 3(x 1 3) 5 3(x 2 3) 218 13. (x 1 3) 1 (x 2 3) 5 3 3 2 14. (x 1 3) 2 (x 2 3) 5 3 impossibility 15. 2(x 1 5) 2 4 5 3(x 1 2) 2 1 1 16. 1 2 (x 2 5) 5 3 2 (x 2 5) impossibility 17. 4(x 2 1) 1 3 5 4x 2 (x 1 1) 0 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 38 Name Class Date Reteaching 1-4 Solving Equations To solve an equation that contains a variable, find all of the values of the variable that make the equation true. Use the equality properties of real numbers and inverse operations to rewrite the equation until the variable is alone on one side of the equation. Whatever remains on the other side of the equation is the solution. Subtraction Property of Equality Addition Property of Equality To isolate w on one side of the equation, add To isolate 5z on one side of the equation, subtract 3 from each side. 2 7 to each side. w 2 7 5 11 2 17 5z 1 3 5 13 2 3 23 5z 5 10 17 w 5 18 2 Division Property of Equality Multiplication Property of Equality To isolate w on one side of the equation, multiply each side by 2. To isolate z on one side of the equation, divide each side by 5. w 5 18 2 32 5z 10 5 5 5 32 z52 w 36 Exercises Solve each equation. 1. y 1 12 5 8 24 2. p 2 9 5 12 21 3. 23 1 r 5 20 23 4. 8 5 15 1 k 27 5. 9q 5 27 3 t 6. 6 5 24 224 d 7. 5 5 2 7 235 8. 49 5 10m 4.9 n 10. 8 1 19 5 3 2128 9. 3g 2 14 5 13 9 c 11. 27 5 25 2 4 2128 12. 12 5 2f 1 9 1.5 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 39 Name Class 1-4 Date Reteaching (continued) Solving Equations To solve an equation for one of its variables, rewrite the equation as an equivalent equation with the specified variable on one side of the equation by itself and an expression not containing that variable on the other side. Problem ax 2 b The equation 2 5 x 1 2b defines a relationship between a, b, and x. What is x in terms of a and b? Use the properties of equality and the properties of real numbers to rewrite the equation as a sequence of equivalent equations. ax 2 b 5 x 1 2b 2 2Q ax 2 b 2 R 5 2(x 1 2b) Multiply each side by 2. ax 2 b 5 2(x 1 2b) Simplify. ax 2 b 5 2x 1 4b Distributive Property ax 2 2x 5 4b 1 b Add and subtract to get terms with x on one side and terms without x on the other side. ax 2 2x 5 5b Simplify. x(a 2 2) 5 5b Distributive Property 5b x5a22 Divide each side by a 2 2. The final form of the equation has x on the left side by itself and an expression not containing x on the right side. Exercises Solve each equation for the indicated variable. 13. 3m 2 n 5 2m 1 n, for m m 5 2n 14. 2(u 1 3v) 5 w 2 5u, for u u 5 w 27 6v b 15. ax 1 b 5 cx 1 d, for x x 5 da 2 2c 1 20 16. k (y 1 3z) 5 4(y 2 5), for y y 5 23kz k24 1 17. 2 r 1 3s 5 1, for r r 5 2 2 6s 5 2 12 2 5g 18. 3 f 1 12 g 5 1 2 f g , for f f 5 8 1 12g 19. 3j 2 4k x1k 3 5 4 , for x x 5 4 j 20. a 2 3y 2 ab 1 4 5 a 1 y, for y y 5 a 1b4b 1 3 b Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 40 Name Class 1-5 Date Additional Vocabulary Support Solving Inequalities is greater than is greater than or equal to is less than is less than or equal to S L R K To write an inequality from a sentence, first identify the operation and then identify the inequality. Example What inequality represents the sentence “6 more than a number is at least 20”? “more than” means addition “is at least” means is greater than or equal to 6 more than a number is at least 20 6 1 x $ 20 Underline the word or words that indicate an operation. 1. the product of 12 and a number 2. 8 less than a number 3. the difference between a number and 24 4. the sum of a number and 7 Circle the word phrase that identifies the inequality to use. Then write the inequality that represents the sentence. 5. The product of 12 and a number is more than 190. 12x S 190 6. 8 less than a number is at least 34. x 2 8 L 34 7. The difference between a number and 24 is no more than 4. x 2 24 K 4 8. The sum of twice a number and 7 is less than 25. 2x 1 7 R 25 Some word phrases are very similar, but have different meanings. Example Does the sentence indicate an operation or an inequality? A number is four 5 greater than 15. x 5 4 Four is 5 greater than a number. 1 15 operation 4 x . inequality Does the sentence indicate an operation or an inequality? 9. 22 is 7 greater than a number. 10. A number is greater than 99. inequality operation 11. A number is 8 less than another number. 12. 50 is less than a number. inequality operation Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 41 Name Class Date Think About a Plan 1-5 Solving Inequalities Your math test scores are 68, 78, 90, and 91. What is the lowest score you can earn on the next test and still achieve an average of at least 85? Understanding the Problem 1. What information do you need to find an average of scores? How do you find an average? The sum of the scores and the total number of scores; divide the sum of the scores by the total number of scores. 5 2. How many scores should you include in the average? ___________ z z z z greater than or equal to what score? 3. You want to achieve an average that is 85 Planning the Solution 4. Assign a variable, x. x 5 the score on the next test 5. Write an expression for the sum of all of the scores, including the next test. 327 1 x 6. Write an expression for the average of all of the scores. 327 1 x 5 7. Write an inequality that can be used to determine the lowest score you can earn on the next test and still achieve an average of at least 85. 327 1 x 5 L 85 Getting an Answer 8. Solve your inequality to find the lowest score you can earn on the next test and still achieve an average of at least 85. What score do you need to earn? at least 98 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 42 Name Class 1-5 Date Practice Form G Solving Inequalities Write the inequality that represents the sentence. 1. Four less than a number is greater than 228. x 2 4 + 228 2. Twice a number is at least 15. 2x # 15 3. A number increased by 7 is less than 5. x 1 7 * 5 4. The quotient of a number and 8 is at most 26. x8 K 26 Solve each inequality. Graph the solution. 5. 3(x 1 1) 1 2 , 11 x * 2 3 2 1 0 1 2 6. 5t 2 2(t 1 2) $ 8 t # 4 1 1 8. 3(7a 3 7. 2f(2y 2 1) 1 yg # 5( y 1 3) y " 17 1 2 5 6 7 2 1) # 2a 1 7 a " 22 10. 22(w 2 7) 1 3 . w 2 1 w * 6 9. 5 2 2(n 1 2) # 4 1 n n # 21 0 4 3 19 20 21 22 23 24 25 14 15 16 17 18 19 20 3 2 1 2 3 3 4 5 6 7 8 9 Solve each problem by writing an inequality. 11. Geometry The length of a rectangular yard is 30 meters. The perimeter is at most 90 meters. Describe the width of the yard. at most 15 m 12. Geometry A piece of rope 20 feet long is cut from a longer piece that is at least 32 feet long. The remainder is cut into four pieces of equal length. Describe the length of each of the four pieces. at least 3 ft 13. A school principal estimates that no more than 6% of this year’s senior class will graduate with honors. If 350 students graduate this year, how many will graduate with honors? no more than 21 students 14. Two sisters drove 144 miles on a camping trip. They averaged at least 32 miles per gallon on the trip. Describe the number of gallons of gas they used. at most 4.5 gal Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 43 Name 1-5 Class Date Practice (continued) Form G Solving Inequalities Is the inequality always, sometimes, or never true? 15. 3(2x 1 1) . 5x 2 (2 2 x) always true 16. 2(x 2 1) $ x 1 7 sometimes true 17. 7x 1 2 # 2(2x 2 4) 1 3x never true 18. 5(x 2 3) , 2(x 2 9) sometimes true Solve each compound inequality. Graph the solution. 19. 3x . 26 and 2x , 6 x + 22 and x * 3 3 2 1 0 1 2 3 20. 4x $ 212 and 7x # 7 x # 23 and x " 1 3 2 1 0 1 2 3 21. 5x . 220 and 8x # 32 x + 24 and x " 4 6 4 2 0 2 4 6 22. 6x , 212 or 5x . 5 x * 22 or x + 1 3 2 1 0 1 2 3 23. 6x # 218 or 2x . 18 x " 23 or x + 9 6 3 3 6 9 12 24. 2x . 3 2 x or 2x , x 2 3 x + 1 or x * 23 4 3 2 1 0 1 2 0 Solve each problem by writing and solving a compound inequality. 25. A student believes she can earn between $5200 and $6250 from her summer job. She knows that she will have to buy four new tires for her car at $90 each. She estimates her other expenses while she is working at $660. How much can the student save from her summer wages? between $4180 and $5230 26. Before a chemist can combine a solution with other liquids in a laboratory, the temperature of the solution must be between 39°C and 52°C. The chemist places the solution in a warmer that raises the temperature 6.5°C per hour. If the temperature is originally 0°C, how long will it take to raise the temperature to the necessary range of values? between 6 and 8 h 27. The Science Club advisor expects that between 42 and 49 students will attend the next Science Club field trip. The school allows $5.50 per student for sandwiches and drinks. What is the advisor’s budget for food for the trip? between $231 and $269.50 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 44 Name Class Date Practice 1-5 Form K Solving Inequalities Write the inequality that represents the sentence. 1. Five less than a number is at least 228. x 2 5 L 228 2. The product of a number and four is at most 210. 4x K 210 3. Six more than a quotient of a number and three is greater than 14. x3 1 6 S 14 Solve each inequality. Graph the solution. 4. 5a 2 10 . 5 a S 3 5. 25 2 2y $ 33 y K 24 To start, add 10 to each side. 7 6 5 4 3 2 1 1 2 3 4 5 6 7 6. 22(n 1 2) 1 6 # 16 n L 27 7. 2(7a 1 1) . 2a 2 10 a S 21 10 9 8 7 6 5 4 4 3 2 1 0 1 2 Solve the following problem by writing an inequality. 8. The width of a rectangle is 4 cm less than the length. The perimeter is at most 48 cm. What are the restrictions on the dimensions of the rectangle? To start, record what you know. width: length 2 4 perimeter: at most 48 cm Describe what you need to find. restrictions on the width and w S 0, l 2 4 S 0 length of the rectangle 2(l 2 4) 1 2l K 48; 4 R l K 14; 0 R w K 10 The length is more than 4 cm and at most 14 cm. The width is more than 0 cm and at most 10 cm. Is the inequality always, sometimes, or never true? 10. 2x 1 8 # 2(x 1 1) never 9. 5(x 2 2) $ 2x 1 1 sometimes 11. 6x 1 1 , 3(2x 2 4) never 12. 2(3x 1 3) . 2(3x 1 1) always Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 45 Name Class Date Practice (continued) 1-5 Form K Solving Inequalities Solve each compound inequality. Graph the solution. 13. 2x . 24 and 4x , 12 x S 22 and x R 3 To start, simplify each inequality. x . 22 and x , 3 2 1 0 1 2 3 4 5 4 3 2 1 0 1 4 5 6 5 4 3 2 1 0 1 5 4 3 2 1 0 1 2 1 3 4 Remember, “and” means that a solution makes BOTH inequalities true. 14. 3x $ 212 and 5x # 5 x L 24 and x K 1 15. 6x . 6 and 9x # 45 x S 1 and x K 5 0 1 2 3 Solve each compound inequality. Graph the solution. 16. 3x , 29 or 8x . 28 x R 23 or x S 21 To start, simplify each inequality. x , 23 or x . 21 Remember, “or” means that a solution makes EITHER inequality true. 17. 7x # 228 or 2x . 22 x K 24 or x S 21 18. 3x . 3 or 5x , 2x 2 3 x S 1 or x R 21 0 1 Write an inequality to represent each sentence. 19. The average of Shondra’s test scores in Physics is between 88 and 93. 88 R x R 93 OR x S 88 and x R 93 20. The Morgans are buying a new house. They want to buy either a house more the 75 years old or a house less than 10 years old. x S 75 or x R 10 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 46 2 Name Class 1-5 Date Standardized Test Prep Solving Inequalities Multiple Choice For Exercises 1–5, choose the correct letter. 1. What is the solution of 4t 2 (3 1 t) # t 1 7? B t # 52 t#5 t#2 t#1 r , 24 r . 24 2. What is the solution of 217 2 2r , 3(r 1 1)? I r.4 r . 220 3 3. Which graph best represents the solution of 4(m 1 4) . m 1 3? A 1 3 2 1 0 1 2 3 1 3 2 1 0 1 2 3 8 9 10 11 12 1 3 2 1 0 1 6 7 4. What is the solution of the compound inequality 4x , 28 or 9x . 18? x , 2 or x . 22 x.2 x , 22 x , 22 or x . 2 2 3 I 5. What is the solution of the compound inequality 22x # 6 and 23x . 227? C x # 23 and x . 9 x $ 23 and x , 9 x $ 3 and x , 29 x # 3 and x . 29 Short Response 6. Geometry The lengths of the sides of a triangle are in the ratio 3 : 4 : 5. Describe the length of the longest side if the perimeter is not more than 72 in. Solve the inequality 3x 1 4x 1 5x " 72 or 12x " 72 or x " 6. The longest side, 5x, is not more than 5 ∙ 6 5 30 in. [2] All student calculations are correct. [1] minor incorrect calculation [0] no answer given 7. Between 8.5% and 9.4% of the city’s population uses the municipal transit system daily. According to the latest census, the city’s population is 785,000. How many people use the transit system daily? [2] between 66,725 and 73,790 people [1] minor incorrect miscalculation of one or both values in the range [0] no answer given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 47 Name Class Date Enrichment 1-5 Solving Inequalities F R I E D R I C H 1 2 3 4 5 6 7 8 9 10 B E S S E L 11 12 13 14 15 16 He was the first person to measure the distance to a star successfully. He also discovered that certain mathematical functions play a key role in models of physical phenomena. These functions were named after him. To find his name, solve each of the following inequalities.Then use the solutions and the table below to determine the position in which to write the associated letter. Solutions Position Solutions Position x1 1 1 x 2 6 x4 11 x2 2 x5 7 x0 12 x 1 3 5 x 2 8 x 2 4 x0 9 4 x 3 x 1 2 all real numbers 5 10 x2 15 x3 16 B 2(x 1 1) 2 (2 2 x) , 12 x R 4 C 3x 1 6 , 16 2 x x R Solutions 5 2 D 3x 1 5 1 3(x 1 5) . 6x 1 15 all real numbers E 8x 2 2 2 6(x 2 3) , 16 x R 0 E 3(x 1 2) 1 2x . 24 x S 22 E (3 1 x) 1 (3 2 3x) , x x S 2 F 3x 2 5 1 (x 1 9) . 8 x S 1 H 5x 2 3(x 2 1) . 3 x S 0 I 5x 2 3 2 3 Q x 1 73 R . 0 x S 5 I 2(x 1 4) 2 (5 2 x) . 0 x S 21 L (x 1 1) 2 (1 2 x) , (x 2 1) 2 (2 2 2x) x S 3 R 7x 2 5 2 (21 2 x) . 0 x S R 2x 1 5 1 2(x 1 5) , 23 x R 2 S 2(x 2 1) , 2 1 2(1 1 7x) x S 2 12 S 3 2 x , 5 1 2(x 1 1) x S 2 43 1 2 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 48 Position 13 14 Name 1-5 Class Date Reteaching Solving Inequalities As with an equation, the solutions of an inequality are numbers that make it true. The procedure for solving a linear inequality is much like the one for solving linear equations. To isolate the variable on one side of the inequality, perform the same algebraic operation on each side of the inequality symbol. The Addition and Subtraction Properties of Inequality state that adding or subtracting the same number from both sides of the inequality does not change the inequality. If a , b, then a 1 c , b 1 c. If a , b, then a 2 c , b 2 c. The Multiplication and Division Properties of Inequality state that multiplying or dividing both sides of the inequality by the same positive number does not change the inequality. If a , b and c . 0, then ac , bc. a If a , b and c . 0, then c , bc . Problem What is the solution of 3(x 1 2) 2 5 # 21 2 x? Graph the solution. Justify each line in the solution by naming one of the properties of inequalities. 3x 1 6 2 5 # 21 2 x 3x 1 1 # 21 2 x 4x 1 1 # 21 4x # 20 x#5 Distributive Property Simplify. Addition Property of Inequality Subtraction Property of Inequality Division Property of Inequality To graph the solution, locate the boundary point. Plot a point at x 5 5. Because the inequality is “less than or equal to,” the boundary point is part of the solution set. Therefore, use a closed dot to graph the boundary point. Shade the number line to the left of the boundary point because the inequality is “less than.” Graph the solution on a number line. 2 3 4 5 6 7 8 Exercises Solve each inequality. Graph the solution. 1. 2x 1 4(x 2 2) . 4 x + 2 3 2 1 0 1 2 2. 4 2 (2x 2 4) $ 5 2 (4x 1 3) x # 23 5 4 3 2 1 3 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 49 0 1 Name 1-5 Class Date Reteaching (continued) Solving Inequalities The procedure for solving an inequality is similar to the procedure for solving an equation but with one important exception. The Multiplication and Division Properties of Equality also state that, when you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol. If a , b and c , 0, then ac . bc. a If a , b and c , 0, then c . bc . Problem What is the solution of 2x 2 3(x 2 1) , x 1 5? Graph the solution. Justify each line in the solution by naming one of the properties of inequalities. 2x 2 3(x 2 1) , x 1 5 2x 2 3x 1 3 , x 1 5 2x 1 3 , x 1 5 22x , 2 x . 21 Distributive Property Simplify. Subtraction Property of Inequality Division Property of Inequality The direction of the inequality changed in the last step because we divided both sides of the inequality by a negative number. Graph the solution on a number line. 1 3 2 1 0 1 2 3 Exercises Solve each inequality. 3. x 2 1 # 24(22 2 x) x # 23 4. 7 2 7(x 2 7) . 24 1 5x x * 5 5. 7(x 1 4) 2 13 $ 12 1 13(3 1 x) x " 26 6. 4x 2 1 , 6x 2 5 x + 2 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 50 Name 1-6 Class Date Additional Vocabulary Support Absolute Value Equations and Inequalities Concept List »x… R 3 »x… S 3 »x… K 3 »x… L 3 »x… 5 3 Choose the concept from the list below that best represents the item in each box. 1. numbers more than 2. numbers three units 3 units away from zero 3. away from zero or more than three units away from 0 »x… S 3 ⫺6 ⫺4 ⫺2 0 2 4 6 »x… R 3 »x… L 3 5. 4. numbers less than 6. numbers 3 units away 3 units away from zero from zero ⫺6 ⫺4 ⫺2 »x… R 3 0 2 4 6 7. 8. numbers three units ⫺6 ⫺4 ⫺2 »x… L 3 »x… 5 3 »x… S 3 0 2 4 6 away from zero or less than three units away from 0 9. ⫺6 ⫺4 ⫺2 »x… K 3 »x… K 3 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 51 0 2 4 6 Name Class 1-6 Date Think About a Plan Absolute Value Equations and Inequalities Write an absolute value inequality to represent the situation. Cooking Suppose you used an oven thermometer while baking and discovered that the oven temperature varied between 15 and 25 degrees from the setting. If your oven is set to 350°, let t be the actual temperature. 1. How do you have to think to solve this problem? If I subtract the set temperature from the real temperature, the result should be between 25° and 5°. 2. Write a compound inequality that represents the actual oven temperature t. 345 K t K 355 3. It often helps to draw a picture. Graph this compound inequality on a number line. 345 350 355 t 4. What is the definition of tolerance? Tolerance is the difference between a desired measurement and its maximum and minimum allowable values. It equals half of the difference between the maximum and minimum values. 5° 5. What is the tolerance of the oven? _________ 6. Use the tolerance to write an inequality without absolute values. 25 K t 2 350 K 5 7. Rewrite the inequality as an absolute value inequality. »t 2 350… K 5 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 52 Name Class 1-6 Date Practice Form G Absolute Value Equations and Inequalities Solve each equation. Check your answers. 1. u23x u 5 18 x 5 6 or x 5 26 2. u 5y u 5 35 y 5 7 or y 5 27 3. u t 1 5 u 5 8 t 5 3 or t 5 213 4. 3 u z 1 7 u 5 12 z 5 23 or z 5 211 5. u 2x 2 1 u 5 5 x 5 3 or x 5 22 6. u 4 2 2y u 1 5 5 9 y 5 4 or y 5 0 Solve each equation. Check for extraneous solutions. 7. u x 1 5 u 5 3x 2 7 x 5 6 8. u 2t 2 3 u 5 3t 2 2 t 5 1 5 9. u 4w 1 3 u 2 2 5 5 w 5 1 or w 5 22 10. 2 u z 1 1 u 2 3 5 z 2 2 z 5 21 Solve each inequality. Graph the solution. 11. 5 u y 1 3 u , 15 26 R y R 0 ⫺8 ⫺6 ⫺4 ⫺2 0 2 12. u 2t 2 3 u # 5 21 K t K 4 ⫺2 ⫺1 4 1 14. 2 u 2w 13. u 4b u 2 3 . 9 b < 23 or b > 3 ⫺9 ⫺6 ⫺3 0 3 6 0 1 2 1 2 3 4 2 1 u 2 3 $ 1 w K 272 or w L ⫺6 ⫺4 ⫺2 9 1 15. 2 u 4x 1 1 u 2 5 # 1 21 K x K 2 ⫺3 ⫺2 ⫺1 0 0 2 4 9 2 6 2 16. u 3z 2 2 u 1 5 . 9 z R 23 or z S 2 ⫺3 ⫺2 ⫺1 3 0 1 2 3 Write each compound inequality as an absolute value inequality. 17. 27.3 # a # 7.3 »a» K 7.3 18. 11 # m # 19 »m 2 15» K 4 19. 28.6 # F # 29.2 »F 2 28.9» K 0.3 20. 0.0015 # t # 0.0018 »t 2 0.00165» K 0.00015 Write an absolute value equation or inequality to describe each graph. 21. ⫺6 ⫺4 ⫺2 0 2 4 22. 6 »x» 5 6 ⫺3 ⫺2 ⫺1 0 1 »x» S 2.5 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 53 2 3 Name 1-6 Class Practice Date Form G (continued) Absolute Value Equations and Inequalities Solve each equation. 23. 3 u 2x 1 5 u 5 9x 2 6 x 5 7 3 24. u 4 2 3m u 5 m 1 10 m 5 7 or m 5 2 2 25. 2 u 4w 2 5 u 5 12w 2 18 w 5 2 3 26. 4 u8t 2 12 u 5 6(t 2 1) t 5 5 4 27. u 5p 1 3 u 2 4 5 2p p 5 13 or p 5 21 28. u 7y 2 3 u 1 1 5 0 no solution Solve each inequality. Graph the solution. 29. 23 u 2t 1 1 u , 9 all real numbers 30. u 22x 1 4 u $ 4 x K 0 or x L 4 0 ⫺3 ⫺2 ⫺1 y 1 2 31. ` 3 ` 2 1 , 2 1 1 32. 7 u 4z 1 5 u 1 2 . 5 ⫺16 ⫺12 ⫺8 ⫺4 0 2 ⫺1 3 211 R y R 7 4 0 1 ⫺9 ⫺6 ⫺3 8 2 0 3 3 4 5 z R 213 2 or z S 4 6 9 Write an absolute value inequality to represent each situation. 33. To become a potential volunteer donor listed on the National Marrow Donor Program registry, a person must be between the ages of 18 and 60. Let a represent the age of a person on the registry. »a 2 39» K 21 34. Two friends are hiking in Death Valley National Park. Their elevation ranges from 228 ft below sea level at Badwater to 690 ft above sea level at Zabriskie »x 2 231» K 459 Point. Let x represent their elevation. 35. The outdoor temperature ranged between 37°F and 62°F in a 24-hour period. Let t represent the temperature during this time period. »t 2 49.5» K 12.5 The diameter of a ball bearing in a wheel assembly must be between 1.758 cm and 1.764 cm. 36. What is the tolerance? 0.003 cm 37. What absolute value inequality represents the diameter of the ball bearing? Let d represent the diameter in cm. »d 2 1.761» R 0.003 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 54 Name Class 1-6 Date Practice Form K Absolute Value Equations and Inequalities Solve each equation. Check your answers. Graph the solution. 1. u 22x u 5 12 x 5 6 or x 5 26 ⫺6 ⫺4 ⫺2 0 2 4 2. u 7y u 5 28 y 5 4 or y 5 24 ⫺6 ⫺4 ⫺2 6 0 2 4 6 Solve each equation. Check your answers. 3. u t 1 7 u 5 1 t 5 26 or t 5 28 t 1 7 5 1 or t 1 7 5 21 To start, rewrite the absolute value equation as two equations. 4. 4 u z 1 1 u 5 24 z 5 5 or z 5 27 5. u 2w 1 1 u 5 5 w 5 2 or w 5 23 6. u 2x 2 2 u 5 4 x 5 3 or x 5 21 7. u 5 2 2y u 1 3 5 8 y 5 0 or y 5 5 Solve each equation. Check for extraneous solutions. 8. u 2z 2 9 u 5 z 2 3 z 5 6 or z 5 4 To start, rewrite as two equations. 2z 2 9 5 z 2 3 or 2z 2 9 5 2(z 2 3) 9. u x 1 6 u 5 2x 2 3 x 5 9 10. u 2t 2 5 u 5 3t 2 10 t 5 5 11. 2 u 4y 1 1 u 5 4y 1 10 y 5 2 or y 5 21 12. u w 1 1 u 2 5 5 2w w 5 22 Write an absolute value equation to describe each graph. 13. ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 2 3 14. 4 Variable may vary. Sample: »x… 5 3 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 Variable may vary. Sample: »x 1 3… 5 1 Is the absolute value equation always, sometimes, or never true? Explain. 15. u w u 5 22 16. u z u 1 1 5 z 1 1 never; the distance between a number w and 0 is never negative sometimes; the equation is true only for z L 0 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 55 Name Class 1-6 Date Practice (continued) Form K Absolute Value Equations and Inequalities Solve each inequality. Graph the solution. 17. 2 u x 1 5 u # 8 29 K x K 21 ⫺9 ⫺8 ⫺7 ⫺6 ⫺5⫺4 ⫺3 ⫺2 ⫺1 ux 1 5u # 4 To start, divide each side by 2. x 1 5 is greater than or equal to 24 and less than or equal to 4. 18. u x 1 1 u 2 3 # 1 25 K x K 3 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 2 0 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 3 0 1 2 3 21. u y 2 3 u 1 2 $ 4 y K 1 or y L 5 0 1 22. u 2t 1 2 u 1 5 # 9 23 K t K 1 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 z R 23 or z S 1 19. u 2z 1 2 u 2 1 . 3 20. 2 u w 1 3 u 2 1 , 1 24 R w R 22 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 24 # x 1 5 # 4 1 2 3 4 5 6 23. u 2s 1 1 u . 3 s R 22 or s S 1 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 1 2 3 Write each compound inequality as an absolute value inequality. 24. 1.2 # a # 2.4 »a 2 1.8… K 0.6 To start, find the tolerance. 2.4 2 1.2 1.2 5 2 5 0.6 2 25. 22 , x , 4 »x 2 1… R 3 26. 1 # m # 2 »m 2 1.5… K 0.5 27. 20 # y # 30 »y 2 25… K 5 28. 23 , t , 17 »t 2 7… R 10 Write an absolute value inequality to represent each situation. 29. In order to enter the kiddie rides at the amusement park, a child must be between the ages of 4 and 10. Let a represent the age of a child who may go on the kiddie rides. »a 2 7… K 3 30. The outdoor temperature ranged between 42°F and 60°F in a 24-hour period. Let t represent the temperature during this time period. »t 2 51… K 9 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 56 Name Class 1-6 Date Standardized Test Prep Absolute Value Equations and Inequalities Multiple Choice For Exercises 1–5, choose the correct letter. 1. What is the solution of u 5t 2 3 u 5 8? C t 5 8 or t 5 28 t 5 11 5 or t 5 21 t 5 1 or t 5 211 5 t 5 5 or t 5 23 8 2. What is the solution of u 3z 2 2 u # 8? F 10 22 # z # 3 10 23 #z#2 8 10 z # 22 or z $ 3 10 z # 2 3 or z $ 2 1 3. What is the solution of 2 u 2x 1 3 u 2 1 . 1? B 7 22 , x , 12 7 x , 22 or x . 12 7 x . 2 or x , 212 7 x , 12 or x . 22 4. Which absolute value inequality is equivalent to the compound inequality 23 # T # 45? I u T 2 11 u # 34 u T 2 45 u # 22 u T 2 24 u # 1 u T 2 34 u # 11 5. Which is the correct graph for the solution of u 2b 1 1 u 2 3 # 2? C ⫺3 ⫺2 ⫺1 0 1 2 3 ⫺3 ⫺2 ⫺1 0 1 2 3 ⫺3 ⫺2 ⫺1 0 1 2 3 ⫺3 ⫺2 ⫺1 0 1 2 3 Short Response 6. An employee’s monthly earnings at an electronics store are based on a salary plus commissions on her sales. Her earnings can range from $2500 to $3200, depending on her commission. Write a compound inequality to describe E, the amount of her monthly earnings. Then rewrite your inequality as an absolute value inequality. 2500 K E K 3200,»E 2 2850… K 350 [2] correct compound inequality and correct absolute value inequality [1] incorrect compound inequality OR incorrect absolute value inequality [0] no answer given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 57 Name Class 1-6 Date Enrichment Absolute Value Equations and Inequalities When is the distance from six to ten less than the distance from one to two? When the distance is traveled on a word ladder! A word ladder is a sequence of words in which only one letter in each word changes. To find a word ladder beginning with the word ONE and ending with the word TWO, solve each of the following absolute value inequalities. Write the solutions in the form a , x , b, where a and b are integers. Associated with each inequality is a pair of letters. Fill in the word ladder by placing the first letter of the pair on the line numbered by a and the second letter on the line numbered by b. O ____ W ____ E ____ 3 2 1 O ____ W ____ L ____ 6 5 4 L O ____ I ____ ____ 9 8 7 I ____ L A ____ ____ 10 11 12 I ____ R A ____ ____ 13 14 15 I ____ R F ____ ____ 16 17 18 F ____ I ____ N ____ 19 20 21 T ____ I ____ N ____ 22 23 24 N T ____ O ____ ____ 27 26 25 O T ____ O ____ ____ 30 28 29 O N E OO u 3x 2 51 u , 39 4 R x R 30 WL 3 , 10 2 u 11 2 2x u 2 R x R 9 RI u 95 2 5x u 1 3 , 23 15 R x R 23 EF u 4x 2 38 u 2 14 , 12 3 R x R 16 IO u x 2 20 u , 9 11 R x R 29 II 2 , 5 2 u x 2 17 u 14 R x R 20 WO u 93 2 6x u , 63 5 R x R 26 LL u 2x 2 18 u 1 4 , 10 6 R x R 12 II u 8x 2 100 u , 36 8 R x R 17 AA 26 , 2u 46 2 4x u 10 R x R 13 RF u 4x 2 74 u 1 7 , 9 18 R x R 19 NN 5 , 20 2 u 225 2 10x u 21 R x R 24 TT 3 1 u 106 2 4x u , 9 25 R x R 28 ON 20 , 70 2 u 5x 2 85 u 7 R x R 27 OT u 92 2 8x u 2 40 , 44 1 R x R 22 T W O Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 58 Name Class Date Reteaching 1-6 Absolute Value Equations and Inequalities Solving absolute value equations require solving two equations separately. Recall that for a real number x, u x u is the distance from zero to x on the number line. The equation u x u 5 p means that either x 5 p or x 5 2p because both are p units from 0. Problem What is the solution set for the equation u 5x 1 1 u 2 3 5 4? The first step in solving an absolute value equation is to isolate the absolute value on one side of the equal sign. u 5x 1 1 u 2 3 5 4 u 5x 1 1 u 2 3 1 3 5 4 1 3 Add 3 to each side. u 5x 1 1 u 5 7 Simplify. Next, rewrite the absolute value as two equations and solve each of them separately. 5x 1 1 5 7 5x 5 6 x5 6 5 or or 5x 1 1 5 27 5x 5 28 Deﬁnition of absolute value or 285 Division Property of Equality x5 Addition Property of Equality Notice that the same operations are performed in the same order on each of the two equations. However, do not try to “simplify” the process by solving a single equation. This leads to errors. The solutions are x 5 65 or x 5 285 . Check each solution in the original equation: Check 6 P5 ? 5 1 1P 2 3 5 4 u6 1 1u 2 3 5 4 4 5 43 8 P 5 ? Q25 R 1 1 P 2 3 5 4 u 28 1 1 u 2 3 5 4 4 5 43 Exercises Solve each absolute value equation. Check your work. 1. u 2x 2 3 u 2 4 5 3 x 5 22 or x 5 5 2. u 3x 2 6 u 1 1 5 13 x 5 6 or x 5 22 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 59 Name Class 1-6 Date Reteaching (continued) Absolute Value Equations and Inequalities To solve an absolute value inequality, keep in mind that u x u is the distance from zero to x on the number line. So, if u x u , p, then x is less than p units from 0, so u x u , p 1 2p , x , p. And, if u x u . p, then x is greater than p units from 0, so u x u . p 1 x , 2p or x . p. In this case, we need to rewrite the absolute value inequality as two separate inequalities. Do not try to combine them into one inequality. Problem What is the solution set for the inequality u 2x 1 3 u . 11? Because the inequality is ., use u x u . p 1 x , 2p or x . p. Begin by rewriting the absolute value as two equations and solve each of them separately. 2x 1 3 , 211 or 2x 1 3 . 11 2x , 214 or 2x . 8 x . 27 or x.4 Rewrite as a compound inequality. Subtract 3 from each side. Divide each side by 2. The solution set is x , 27 or x . 4. Exercises Complete the steps to solve the inequality P 2x 2 4 P K 3 . 3. 4. 5. 23 u 1 u 2 u # x 224 # # x 2 # # x # 3 u 7 u 14 u Rewrite as a compound inequality. u Add 4 to each part. u Multiply each part by 2 . 6. What is the solution? 2 K x K 14 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 60 Name Class Date Chapter 1 Quiz 1 Form G Lessons 1-1 through 1-3 Do you know HOW? 1. Describe a rule for the pattern. Answers may vary. Sample: Stack one triangle on top of the leftmost triangle; add two triangles to the right of the rightmost triangle. 3 18 2. Name all the integers in the list: 0, 22, 5 , p, !7 , 121, !9 ,2 6 . 0, 22, 121, !9,218 6 Name the property of real numbers illustrated by each equation. 4. 3 1 p 5 p 1 3 Commutative Property of Addition 3. 2(3x 2 y) 5 6x 2 2y Distributive Property Write an algebraic expression to model the word phrase. 5. the sum of g and the quotient of 3 and h g1 6. six times the difference of x and 22 6(x 2 22) 3 h Evaluate the expression for the given value of the variable. 7. 12x 2 9(x 2 1); x 5 2 15 8. t(2t 1 3) 1 t 1 6 ; t 5 22 2 Do you UNDERSTAND? 9. Writing How can you use a graph to find a pattern? Answers may vary. Sample: Choose some points on the graph; use these points to make a table of input and output values; look for a pattern in the process column. 10. Vocabulary What is another name for an additive inverse? opposite 11. Reasoning Is there a Multiplication Property of Closure that applies to irrational numbers? Justify your answer. No. Answers may vary. Sample: !2 ? !2 5 2 , which is not irrational. 12. Compare and Contrast What is the difference between simplifying an expression and evaluating an expression? Answers may vary. Sample: Simplifying an expression is rewriting it using the properties of real numbers and combining like terms, resulting in a simpler expression. Evaluating an expression is substituting values for the variables, resulting in a numeric value. Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 61 Name Class Date Chapter 1 Quiz 2 Form G Lessons 1-4 through 1-6 Do you know HOW? Solve each equation. Check your answer. 1 2. 2 (3 2 w) 5 w 1 9 w 5 25 1. 3(t 2 2) 5 2t 1 11 t 5 17 3. Write an equation to solve the problem. Then solve the equation. Maria is 3 years older than her brother, Luis. Luis is 2 years older than their younger sister, Karla. The sum of their ages is 55. How old are the three siblings? Let L be Luis’ age: (L 1 3) 1 L 1 (L 2 2) 5 55; Maria is 21, Luis is 18, Karla is 16 Solve each inequality. Graph the solution. 5. 26a # 18 and 5 2 3a . 2 23 " a * 1 4. 2(3 2 2n) , 2 2 (n 1 5) n + 3 0 1 2 3 4 5 ⫺4 ⫺3 ⫺2 ⫺1 6 0 1 2 6. The city’s registrar of voters estimates that not less than 48% and no more than 63% of voters will vote in the next mayoral election. Currently, there are 238,000 registered voters in the city. How many people will vote in the next election? between 114,240 and 149,940, inclusive Solve each equation. Check for extraneous solutions. 7. |2p 2 7| 1 6 5 11 p 5 1 or p 5 6 9. Solve the inequality 8. |m 2 2| 5 2m 1 5 m 5 21 u t 22 3 u 1 1 . 4 and graph the solution. t * 23 or t + 9 ⫺6 ⫺3 0 3 6 9 12 10. In a chemical process, between 77 g and 85 g of carbon is added to a mixture. Let C be the amount of carbon. Write an absolute value inequality describing the amount of carbon added. |C 2 81| * 4 Do you UNDERSTAND? 11. Writing Suppose the sum of three consecutive even integers is given. How can you find the three numbers? Answers may vary. Sample: Let n be the ﬁrst integer. Even integers differ by 2, so the next one is n 1 2 and the third is n 1 4. Therefore, n 1 (n 1 2) 1 (n 1 4) 5 the given sum. Solve the equation for n to get the ﬁrst number. The other two numbers are n 1 2 and n 1 4. 12. Compare and Contrast How do the solutions to |x| # 1, |x| # 0, and |x| # 21 differ? Answers may vary. Sample: |x| " 1 has an inﬁnite number of solutions, 21 " x " 1; |x| " 0 has a single unique solution, x 5 0; and |x| " 21 has no solution. Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 62 Name Class Date Chapter 1 Test Form G Do you know HOW? Identify a pattern and find the next three numbers in the pattern. 1. 24, 21, 2, 5, . . . 2. 3, 21, 147, 1029, . . . Each term is 3 more than the previous Each term is 7 times more than the term; 8, 11, 14 previous term; 7203; 50,421; 352,947 3. Justify each step by naming the property used. a. 4 Q 23 ? 14 R 5 4 Q 14 ? (23) R Commutative Property of Multiplication b. 1 5 Q 4 ? 4 R ? (23) Associative Property of Multiplication c. 5 1 ? (23) Inverse Property of Multiplication d. 5 23 Identity Property of Multiplication Evaluate the expression for the given value of the variable. 4. 2a2 1 4a 2 17; a 5 5 222 5. 6(s 2 2) 2 4(s 1 1) ; s 5 3 21 3s 1 1 6. The expression 19.95 1 0.02x models the daily cost in dollars of renting a car. In the expression, x represents the number of miles the car is driven. What is the cost of renting a car for a day when the car is driven 50 miles? $20.95 Solve each equation. 7. 3r 1 3.7 5 5r 2 2.5 3.1 8. 3(5t 1 2) 5 36 2 Solve each equation for x. State any restrictions on the variables. 3t 9. tx 2 ux 5 3t x 5 t 2 u , t u u 10. x23 6 1 3 5 a x 5 6a 2 15 Solve each formula for the indicated variable. p 12. P 5 2/ 1 2w, for / < 5 2 2 w or < 5 P 22 2w 1 11. R 5 2 (r1 1 r2), for r2 r2 5 2R 2 r1 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 63 Name Class Date Chapter 1 Test (continued) Form G Write an equation and solve the problem. 13. Two buses leave Dallas at the same time and travel in opposite directions. One bus averages 58 mi/h, and the other bus averages 52 mi/h. When will they be 363 mi apart? 3 h 18 min later Solve each inequality. Graph the solution. 3 15. 4a . 3(a 1 1) 2 Q 7 2 2a R a R 8 14. 3m 1 7 $ 4 m L 21 ⫺3 ⫺2 ⫺1 0 1 2 3 5 6 7 8 9 10 11 Solve each compound inequality. Graph the solutions. 16. 3x 2 1 # 5 or 2x 2 4 $ x 17. 23t # 12 and 22t . 26 x K 2 or x L 4 0 1 2 3 24 K t R 3 4 5 ⫺6 ⫺4 ⫺2 6 0 2 4 6 Solve each equation. Check for extraneous solutions. 18. u 2x 1 3 u 5 5 x 5 1 or x 5 24 19. u x 1 6 u 5 2x x 5 6 20. The temperature T of a refrigerator is at least 35°F and at most 41°F. Write a compound inequality and an absolute value inequality for the temperature of the refrigerator. 35 " T " 41; |T 2 38| " 3 Do you UNDERSTAND? 21. Open-Ended There is no Closure Property of Division that applies to integers. For example, 2 4 3 is not an integer. What is another example of a set of real numbers that does not have a Closure Property for one of the basic operations? Give a specific example to illustrate your claim. Answers may vary. Sample: Closure Property of Multiplication for negative numbers; (22) ? (24) 5 8, which is not a negative number. 22. Reasoning Explain in words why |x| , 0 has no solution. Answers may vary. Sample: |x| * 0 represents the set of numbers x that are fewer than 0 units from 0 on the number line. Since there are no numbers less than 0 units from 0 on the number line, the inequality |x| * 0 has no solution. 23. Writing Explain the difference between the solution(s) to an equation and the solutions to an inequality. Answers may vary. Sample: Equations generally have a ﬁnite (or countable) number of solutions, whereas inequalities typically have an inﬁnite number of solutions. Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 64 Name Class Date Chapter 1 Quiz 1 Form K Lessons 1–1 through 1–3 Do you know HOW? 1. Describe a rule for the pattern. Answers may vary. Sample: Start with a column of two circles. Add one circle to the right of the top row; add two circles to the right of the bottom row. 7 6 2. Identify the integers in the list: 21, 9, 0, p, !25, 81, !5, 22. 21, 0, !25, 81, 2 62 Name the property of real numbers illustrated by each equation. 3. 3(2 ? 5) 5 (3 ? 2) ? 5 4. 5(x 1 2) 5 5 ? x 1 5 ? 2 Associative Property of Multiplication Distributive Property Write an algebraic expression to model the word phrase. 5. the sum of y and the product of 7 and x y 1 7x 6. eight more than the quotient of t and 2 2t 1 8 Evaluate the expression for the given value of the variable. 7. 2b 1 6(b 2 4); b 5 8 40 8. x 1 2(x 2 1); x 5 24 214 Do you UNDERSTAND? 9. Writing Give an example of a number that is an irrational number. Explain why it is irrational. Answers may vary. Sample: π ; π is an irrational number because 3.141592 . . . neither terminates nor repeats; π cannot be written as a quotient of integers. Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 65 Name Class Date Chapter 1 Quiz 2 Form K Lessons 1–4 through 1–6 Do you know HOW? Solve each equation. Check your answer. 1. 2(x 2 1) 5 4x 2 12 5 2. 3(2 2 2y) 5 y 2 8 2 3. Write an equation to solve the problem. Then solve the equation. Brandon has 3 more dollars than Mark. Ashley has 10 dollars less than Mark. They have 35 dollars together. How many dollars does each person have? Variable may vary. Sample: Let d be the number of dollars Mark has: (d 1 3) 1 d 1 (d 2 10) 5 35; Brandon has $17, Mark has $14, Ashley has $4. Solve each inequality. Graph the solution. 4. 18 2 2a $ 26 a K 24 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 5. 5x , 215 or 3x . 23 x R 23 or x S 21 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 0 1 Solve each equation. Check for extraneous solutions. 7 7. u 2x 1 2 u 5 x 1 5 x 5 3 or x 5 23 6. 3 u z 1 1 u 5 21 z 5 6 or z 5 28 8. Solve the inequality 3u w 1 1 u 2 2 , 1 and graph the solution. 22 R w R 0 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 9. Carla buys a package of flower seeds to plant in her garden. The flower seed package advertizes there are between 75 and 83 seeds in each package. Let s be the number of seeds in the package. Write an absolute value inequality describing the number of seeds in the package. »s 2 79… K 4 Do you UNDERSTAND? 10. Writing Write an inequality that has no solution. Explain why it does not have a solution. Answers may vary. Sample: 6x 1 1 R 3(2x 2 4); Using the Distributive Property, 6x 1 1 R 6x 2 12; subtract 6x from each side; 1 R 212. Since 1 R 212 is never true, the inequality has no solution. Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 66 Name Class Date Chapter 1 Test Form K Do you know HOW? Identify a pattern and find the next three numbers in the pattern. 1. 25, 21, 3, 7, c Each term is 4 more than the previous term; 11, 15, 19 2. 64, 32, 16, 8, c Each term is half of the previous term; 4, 2, 1 3. What properties of real numbers are illustrated by each equation below? a. 28 1 3 5 3 1 (28) Commutative Property of Addition b. 4 1 (24) 5 0 Inverse Property of Addition c. 2(8 1 t) 5 2 ? 8 1 2 ? t Distributive Property 7 8 d. 8 ? 7 5 1 Inverse Property of Multiplication Evaluate the expression for the given value of the variable. 4. a2 2 2(a 1 1); a 5 3 1 5. 5(2s 2 1) 2 3(s 1 2); s 5 4 17 6. The expression 15 1 5x models the daily cost in dollars of renting scuba gear from the water sports store. In the expression, x represents the number of hours the scuba gear is used. What is the cost of renting scuba gear for a day when the gear is used for 3 hours? $30 Solve each equation. 7. 2r 1 2 5 3r 2 5 7 8. 8(t 1 1) 5 64 7 Solve each equation for x. State any restrictions on the variables. 9. xt 2 1 135a 8 10. x 5 8a 2t 23 ; t u 0 6x 2 1 5y 5 x5 5y 1 1 6 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 67 Name Class Date Chapter 1 Test (continued) Form K Write an equation and solve the problem. 11. Two buses leave Columbus, Ohio at the same time and travel in opposite directions. One bus averages 55 mi/h and the other bus averages 48 mi/h. When will they be 618 mi apart? 6 h Solve each inequality. Graph the solution. 13. 2a 1 5 , 6a 1 1 a S 1 12. 2n 1 1 $ 7 n L 3 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Solve each compound inequality. Graph the solutions. 14. 3x # 26 or 2x 1 1 $ 3 x K 22 or x L 115. 22t 1 2 , 4 and 2t , 6 21 R t R 3 ⫺3 ⫺2 ⫺1 0 1 2 ⫺3 ⫺2 ⫺1 3 0 1 2 3 Solve each equation. Check for extraneous solutions. 16. u 3x 1 3 u 5 18 x 5 5 or x 5 27 17. u b 1 2 u 5 2b b 5 2 18. The weatherman announced that the temperature T over the next few weeks will be at least 648F and at most 788F. Write an absolute value inequality for the temperature over the next few weeks. »T 2 71… K 7 Do you UNDERSTAND? 19. What is another name for the multiplicative inverse? reciprocal 20. Reasoning Explain in words why 2 u x u , 24 has no solution. Answers may vary. Sample: Dividing both sides by 2 gives |x| R 22. The absolute value of any number must be nonnegative, so the inequality has no solution. 21. Open-Ended What is the difference between simplifying an expression and evaluating an expression? Answers may vary. Sample: Simplifying an expression is rewriting it using the properties of real numbers and combining like terms, resulting in a simpler expression. Evaluating an expression is substituting values for the variables, resulting in a numerical value. Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 68 Name Class Date Chapter 1 Performance Tasks Give complete answers. Task 1 Write an algebraic expression that requires each of the following properties of real numbers to simplify, in the order given: the Distributive Property, the Associative Property of Addition, the Commutative Property of Addition, and the Distributive Property. Simplify the expression showing the use of each property. Answers may vary. Check students’ work. [4] Student writes an algebraic expression satisfying the conditions and correctly simpliﬁes the expression, labeling each step with the appropriate property. [3] Student writes an algebraic expression satisfying the conditions. Simpliﬁcation contains only minor errors. [2] Student writes an algebraic expression that satisﬁes only two or three of the conditions. Simpliﬁcation contains errors. [1] Student writes an algebraic expression that satisﬁes one or none of the conditions. Simpliﬁcation contains signiﬁcant errors or is missing. [0] Response is missing or inappropriate. Task 2 a. Explain how you determine the total number of points scored by a basketball player who makes free-throws (worth 1 point), two-point goals (2 points), and three-point goals (3 points) in a game. b. Use your method from part a to find the total points scored by each of the following three players. Free-throws 2-Point Goals 3-Point Goals Player A 3 5 1 Player B 0 7 3 Player C 2 6 0 a. The total number of points is the sum of 1 times the number of free-throws, 2 times the number of two-point goals, and 3 times the number of three-point goals. b. A: 16 points, B: 23 points, C: 14 points [4] Student gives a detailed explanation of how to ﬁnd the total. Steps in ﬁnding the total for three players are correct. [3] Student gives an adequate explanation of how to ﬁnd the total. Steps in ﬁnding the total for three players contain minor errors. [2] Student does not fully explain how to ﬁnd the total. Steps in ﬁnding the total for three players contain errors. [1] Student does not correctly or completely explain how to ﬁnd the total. Steps in ﬁnding the total for three players contain major errors. [0] Response is missing or inappropriate. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 69 Name Class Date Chapter 1 Performance Tasks (continued) Task 3 Explain how the properties of inequalities differ from the properties of equality and how the solutions of an inequality differ from the solutions of an equation. Use the following equation and inequality as part of your explanation. 25x 5 10 25x . 10 Answers may vary. Sample: When multiplying or dividing both sides of an inequality by a negative value, the inequality symbol must reverse direction. The solution of the inequality is x * 22, whereas the solution of the equation is x 5 22. The solutions of the inequality do not include the solution of the equation, but consist of all numbers less than it. [4] Student gives speciﬁc details of the differences and effectively uses the examples to illustrate these differences. Steps in solving the equation and inequality are correct. [3] Student gives adequate details of the differences and makes reference to the examples. Steps in solving the equation and inequality contain minor errors. [2] Student solves the equation and the inequality but does not explain the differences. Steps in solving the equation and inequality contain errors. [1] Student does not correctly or completely solve the equation and inequality. Steps in solving the equation and inequality contain major errors. [0] Response is missing or inappropriate. Task 4 a. Write an inequality using an absolute value that can be rewritten as a compound inequality with or. b. Solve and graph the inequality in part a. c. Write an inequality using an absolute value that can be rewritten as a compound inequality with and. d. Solve and graph the inequality in part c. a. Answers may vary. Sample: |x| + 2 b. Answers may vary. Sample: x + 2 or x * 22; ⫺6 ⫺4 ⫺2 0 2 4 6 c. Answers may vary. Sample: |x| * 2 d. Answers may vary. Sample: 22 * x * 2; ⫺6 ⫺4 ⫺2 0 2 4 6 [4] Student writes and solves inequalities using an absolute value with no mistakes. Graphs are accurate and detailed. [3] Student correctly creates inequalities using an absolute value. Graphs or solutions contain minor errors. [2] Student makes signiﬁcant errors in writing or solving the inequalities. Graphs contain major errors. [1] Student does not write or solve all inequalities. Graph is incomplete or missing. [0] Response is missing or inappropriate. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 70 Name Class Date Chapter 1 Cumulative Review Multiple Choice For Exercises 1–10, choose the correct letter. 1. What is the next number in the pattern? 1, 2, 4, 7, 11, … D 12 13 15 16 2. Which of the following numbers is an integer but not a natural number? G 223 0 !9 7 3. Which of the following lists of numbers is ordered from least to greatest? C 3 3 3 0, 21, 22, 23, 24 21, 22, 0, 1, 2 0, 1, !2, 2, !3 0, 21, 1, 22, 2 4. Which of the following expressions is not equivalent to 4(1 2 2x) 1 2(5 2 x)? H 14 2 10x 4 2 8x 1 10 2 2x 4 2 8x 1 10 2 x 4 1 10(1 2 x) 5. What is the value of |x 1 3| 1 5x 2 7 for x 5 29? A 246 258 64 240 6. The expression 20,000 2 1250t models the value, in dollars, of a piece of equipment t years after purchase. What is the value of the equipment after 8 years? I $15,000 $0 $150,000 $10,000 7. What is the solution of 4[x 2 (3 2 2x)] 1 5 5 3(x 1 11)? B 31 40 2 3 9 5 8. Which number is not a solution of the inequality 227 # 3(1 2 2x) # 3? G 1 0 21 4 2 9. Which number is not a solution of the equation |x 1 5| 5 x 1 5? D 5 0 25 210 10. Which of the following equations have more than one solution? F II. 2 Q x 1 12 R 5 2x 1 1 III. |x 12| 5 21 I. |x 2 3| 5 4 I and II only I and III only IV. II and III only Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 71 3x 1 5 5 14 I, II, III, and IV Name Class Date Chapter 1 Cumulative Review (continued) Short Response Number of Teachers 11. Derek has noticed there are fewer students per class at his friend’s private school than at his public school. He talks to the principal of the private school and learns that the number of students is the same in every class. Derek surveys the classes and makes the graph at the right, comparing the number of students and the number of teachers. a. How many teachers are required for 85 students at the private school? 5 b. How many teachers are required for 153 students? 9 4 3 2 1 17 34 51 68 85 Number of Students √ 11 5 13 1 12. Graph the numbers 23, 53, and !11 on a number line. 1 13. What are the opposite and the reciprocal of 224 ? 5 ⫺9 ⫺6 ⫺3 0 3 6 9 opposite: 214; reciprocal: 249 14. Write an equation to solve the problem. Find three consecutive even numbers whose sum is 144. n 1 (n 1 2) 1 (n 1 4) 5 144 15. Solve ax 2 3x 1 5 5 a 1 b for x. State any restrictions on the variables. x 5 a 1 b25 a23 , au3 Solve each inequality. Graph the solution. 1 16. 3(t 2 2) 1 5 # 4t 1 2 t L 232 ⫺3 ⫺2 ⫺1 0 1 2 18. u 3x 2 1 u , 7 17. 2x 1 5 , 3 or 6 2 x # 5 x R 21 or x L 1 3 ⫺3 ⫺2 ⫺1 0 1 2 22 R x R 83 3 ⫺3 ⫺2 ⫺1 0 1 2 19. Write the compound inequality 3.1 # m # 4.7 as an absolute value inequality. |m 2 3.9| K 0.8 Extended Response 1 20. Writing What is the value of the expression 0 ? 0? Explain. It is undeﬁned; because division by 0 is undeﬁned, the entire expression is undeﬁned. [4] appropriate methods and explanation with no computational errors [3] appropriate methods and explanation, but with one computational error [2] incorrect value OR correct value without explanation [1] correct value, without work shown or explanation [0] answer missing or no attempt made Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 72 3 TEACHER INSTRUCTIONS Chapter 1 Project Teacher Notes: Buy the Hour About the Project The Chapter Project gives students an opportunity to use expressions, equations, inequalities, and graphs to model real-life situations. Students write and simplify expressions by using the Distributive Property and combining like terms, evaluate expressions, and write and solve equations and inequalities in one variable. Introducing the Project • Encourage students to keep all project-related materials in a separate folder. • Ask students what the term minimum wage means. • Have students explain the similarities and differences among the terms expression, equation, and inequality. Remind students that it is important to define the variable when writing expressions, equations, and inequalities. Activity 1: Researching Students research federal and state minimum wages. Activity 2: Modeling Students write and evaluate expressions modeling amounts of money earned based on minimum wages found in Activity 1. Activity 3: Solving Students write and solve equations and inequalities that model relationships between amounts of money earned. Students also graph the solutions of their inequalities. Finishing the Project You may wish to plan a project day on which students share their completed projects. Encourage students to explain their processes as well as their results. • Have students review their methods for writing and evaluating expressions; for writing and solving equations; and for writing, solving, and graphing inequalities. • Ask groups to share their insights that resulted from completing the project, such as shortcuts they found for solving equations and inequalities or for researching data. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 73 Name Class Date Chapter 1 Project: Buy the Hour Beginning the Chapter Project Have you had your first job yet? If so, you were probably paid an hourly wage. The amount of money you earned for each hour you worked may have been the minimum wage. This amount, set by the U.S. Department of Labor, is the minimum amount for one hour of work an employer is allowed to pay to employees who meet certain specific criteria. Each state may set its own minimum wage, but where federal and state laws set different rates, the employer is required to pay the greater of the two amounts to all employees to whom the conditions of the federal law apply.* In this project, you will write expressions that model amounts of money earned. You will write equations and inequalities to determine the number of hours that must be worked to satisfy certain conditions. You will also research the current federal and state minimum wage laws. Activities Activity 1: Researching Research the current federal minimum wage. Then find out whether the state in which you live has set its own minimum wage. If so, what is that wage? Select a state other than the state in which you live. Research the minimum wage for that state. You might find it helpful to contact the U.S. Department of Labor and state labor commissioners, or to use the Internet to find this data. Check students’ work. Activity 2: Modeling Suppose you earn the minimum wage determined in Activity 1 for the state other than your own. • Suppose that next week you plan to work h hours. Write an expression that models the amount of money you will earn. Answers may vary. Sample (based on minimum wage of $7.25 effective July 24, 2009): 7.25h • Suppose that your friend earns the same hourly wage that you earn, but works in a job for which he receives tips. Write an expression that models your friend’s total earnings for a week during which he works n hours and receives $15 in tips. Then, evaluate the expression for n 5 10 and explain what this number means. 7.25n 1 15, $87.50, the total earned when your friend works 10 h • Write an expression that models the sum of your earnings for 3 weeks and your friend’s earnings for 2 weeks if you each work r hours per week and your friend receives $15 in tips per week. Simplify the expression. 3(7.25r) 1 2(7.25r 1 15) 5 36.25r 1 30 • Write an expression that models the difference between your earnings and your friend’s earnings for a week during which you work h hours, your friend works n hours, and your friend earns t dollars in tips. (Hint: Be sure to consider the fact that you do not know who earns more money!) |7.25h 2 (7.25n 1 t)| *Source: http://www.dol.gov/dol/topic/wages/minimumwage.htm Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 74 Name Class Date Chapter 1 Project: Buy the Hour (continued) Activity 3: Solving Round numbers of hours to the nearest tenth if necessary. • Suppose that last week your employer gave you a $.50/h raise and a $20 bonus as a reward for good work. You earned a total of $80 for the week. Let x represent the number of hours you worked that week. Write an equation to model this situation. Then solve your equation and explain the meaning of your solution. Answers may vary. Sample (based on minimum wage of $7.25 effective July 24, 2009): 7.75x 1 20 5 80; 7.7; you worked about 7.7 h last week to earn $80. • Suppose your friend (still earning minimum wage) receives $20 in tips, and that you (earning $.50/h more than your friend) have earned the same amount of money at the end of a week during which you worked the same number of hours as your friend. Write an equation to model this situation. Then solve your equation and explain the meaning of your solution. 7.25h 1 20 5 7.75h, where h is the number of hours each worked; 40; each worked 40 h. • Suppose that your friend wants to earn at least $95 next week and he expects to earn $15 in tips. Write an inequality that models this situation. Then solve and graph your inequality. Explain the meaning of your solution. 7.25x 1 15 # 95; x # 11.0; your friend must work at least 11.0 h to earn at least $95 next week. Finishing the Project 8 9 10 11 12 13 14 The answers to the activities should help you to complete your project. You should prepare a presentation for the class describing your results. Your presentation should include the data you researched; the expressions, equations, and inequality you used to model the given situations; and the graph of your inequality. Reﬂect and Revise Ask a classmate to review your project. After you have reviewed each other’s presentations, decide if your work is clear, complete, and convincing. If needed, make changes to improve your presentation. Extending the Project Research the minimum wages set by other states. If they differ from the minimum wage of your state, determine possible factors that might contribute to the differences. Find out what conditions might exist that would allow an employer to pay an employee less than the federal minimum wage. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 75 Name Class Date Chapter 1 Project Manager: Buy the Hour Getting Started Read the project. As you work on the project, you will need a calculator and materials on which you can record your results and make calculations. Keep all of your work for the project in a folder. Checklist Suggestions ☐ Activity 1: researching minimum wages ☐ Select a state in which you are interested. ☐ Activity 2: writing algebraic expressions ☐ Substitute reasonable values for the variables to determine if the expressions make sense. ☐ Activity 3: writing and solving equations and inequalities ☐ Check that your answers are reasonable. ☐ algebraic models ☐ Have you defined the variables in your expressions, equations, and inequality? How does the graph of an equation differ from the graph of an inequality? What does this mean in terms of your solution? Scoring Rubric 4 The expressions, the equations, and the inequality are correct. The graph and all calculations are accurate. Explanations are thorough and well thought out. The presentation is clear and complete. 3 The expressions, the equations, and the inequality have minor errors. The graph and calculations are mostly correct. The explanations and presentation lack detail or contain small errors. 2 The expressions, the equations, and the inequality have major errors. The graph and calculations contain minor errors. The explanations and presentation contain minor inaccuracies. 1 The expressions, the equations, and the inequality are not correct. The graph is not accurate. Calculations contain major errors or are incomplete. The explanations and presentation are inaccurate or incomplete. 0 Major elements of the project are incomplete or missing. Your Evaluation of Project Evaluate your work, based on the Scoring Rubric. Teacher’s Evaluation of the Project Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 76