Lesson 6.1 Skills Practice

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1 Lesson 6.1 Skills Practice Name Date What s It Really Saying? Evaluating Algebraic Expressions Vocabulary Write an example for each term. 1. variable The variable h represents an unknown number of hours. 2. algebraic expression 35h 3. evaluate an algebraic expression When h 5 5, 35h 5 35(5) Problem Set Define a variable and write an algebraic expression for each problem. Evaluate the expression for the given values. 1. The charge for ice skating is $3 for the skate rental and $2 per hour to skate. How much will you pay if you skate for: a. 2 hours? b. 4 hours? c hours? Let h represent the number of hours you skate; 3 1 2h a. You will pay $7 to skate for 2 hours (2) 5 7 b. You will pay $11 to skate for 4 hours (4) 5 11 c. You will pay $10 to skate for hours ( ) 5 10 Chapter 6 Skills Practice 555

2 Lesson 6.1 Skills Practice page 2 2. A birthday party at the skating rink costs $45 to reserve a party area and $4.50 per guest for skating and skate rental. How much will a party cost if you invite: a. 5 guests? b. 9 guests? c. 25 guests? Let g represent the number of guest invited; g a. The party will cost $67.50 for 5 guests (5) b. The party will cost $85.50 for 9 guests (9) c. The party will cost $ for 25 guests (25) You have $15 to spend at the snack bar. All of the snacks at the snack bar cost $1.75. How much money will you have left if you buy: a. 3 snacks? b. 5 snacks? c. 8 snacks? Let s represent the number snacks you bought; s a. You will have $9.75 left if you buy 3 snacks (3) b. You will have $6.25 left if you buy 5 snacks (5) c. You will have $1 left if you buy 8 snacks (8) Chapter 6 Skills Practice

3 Lesson 6.1 Skills Practice page 3 Name Date 4. The zamboni can resurface 1050 square feet per minute. How many minutes will it take the zamboni to resurface the entire rink if its dimensions are: a. 70 ft ft? b. 84 ft ft? c. 90 ft ft? Let w represent the width and l represent the length of the rink; lw 1050 a. It will take 10 minutes to resurface a 70 ft ft rink. 70? b. It will take 8 minutes to resurface a 84 ft ft rink. 84? c. It will take 15 minutes to resurface a 90 ft ft rink. 90? The skating rink is running a promotion on skating lessons. For every ten lessons you take, you get one free lesson. If you have already taken 4 lessons, how many free lessons will you get if you take: a. 6 more lessons? b. 16 more lessons? c. 26 more lessons? Let a represent the number of additional lessons you take; a a. You will get 1 free lesson with 6 more lessons b. You will get 2 free lessons with 16 more lessons c. You will get 3 free lessons with 26 more lessons Chapter 6 Skills Practice 557

4 Lesson 6.1 Skills Practice page 4 6. One lap around the skating rink is about 400 feet and the length is 150 feet. How far will you skate if you skate: a. the length 3 times plus 20 laps? b. the length 5 times plus 35 laps? c. the length 2 times plus 48 laps? Let m represent the number of times you skate the length and n represent the number of times you skate a lap; 150m 1 400n a. You will skate 8450 ft from 3 lengths and 20 laps. 150(3) 1 400(20) b. You will skate 14,750 ft from 5 lengths and 35 laps. 150(5) 1 400(35) 5 14,750 c. You will skate 19,500 ft from 2 lengths and 48 laps. 150(2) 1 400(48) 5 19,500 Evaluate each algebraic expression x a. for x x 5 20(2) 5 40 b. for x x 5 20(28) c. for x x 5 20(0) c 1 17 a. for c 5 5 3c (5) b. for c c (22) c. for c c (215) Chapter 6 Skills Practice

5 Lesson 6.1 Skills Practice page 5 Name Date p a. for p p (4) 5 28 b. for p p (9) c. for p p (23) w a. for w w (12) b. for w w (21.5) c. for w w (5.3) (22k) a. for k (22k) (22)(3) 5 40 b. for k (22k) (22)(23) 5 0 c. for k (22k) (22)(22) 5 50 Chapter 6 Skills Practice 559

6 Lesson 6.1 Skills Practice page n a. for n n (210) b. for n 5 25 n (25) c. for n 5 3 n (3) d a. for d d (2) b. for d d (22) c. for d d (4) q a. for q q (0) b. for q q (10) c. for q q (23) Chapter 6 Skills Practice

7 Lesson 6.1 Skills Practice page 7 Name Date Complete each table. 15. b 3b t 2(t 2 2 6) v v f f 1 3f Chapter 6 Skills Practice 561

8 Lesson 6.1 Skills Practice page s s z z Evaluate each algebraic expression for the given quantity x 2 6.8x, x x 2 6.8x 5 4.9(1.5) 2 6.8(1.5) x 1 8.1x, x x 1 8.1x (5.2) 1 8.1(5.2) Chapter 6 Skills Practice

9 Lesson 6.1 Skills Practice page 9 Name Date x 1 1.4x, x x 1 1.4x (29.3) 1 1.4(29.3) x x, x x x ( 2 5 ) ) ( ( 2 3 ( 2 5 ) 15 5 ) x x, x x x ( ( ) ) ( ( ) 4 ) x x, x x x ( 3 5 ( 3 8 ) ) 5 ( 3 5 ( ) ) Chapter 6 Skills Practice 563

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11 Lesson 6.2 Skills Practice Name Date Express Math Simplifying Expressions Using Distributive Properties Vocabulary Match each property to the correct example. 1. Distributive Property of Multiplication c a. 5(10 2 3) 5 5(10) 2 5(3) over Addition 2. Distributive Property of Multiplication a b over Subtraction 3. Distributive Property of Division b c. 3(6 1 2) 5 3(6) 1 3(2) over Addition 4. Distributive Property of Division d d over Subtraction Problem Set Draw a model for each expression and calculate or simplify. 1. 7(54) 2. 26(870) (2420) (560) 4. 3(x 2 11) x x x 2 33 Chapter 6 Skills Practice 565

12 Lesson 6.2 Skills Practice page (2x 1 5) 6. a x 5 a x x a a (24b 2 9) 8. 25(12x 2 7) 24b b x x b x a b a b a a b b 2 5 Use the Distributive Property to rewrite each expression in its equivalent form (x 1 3) 4x y(4 2 y) 228y 1 7y x(3x 1 5y 2 4) 18x xy 2 24x x x a a n 1 300m n 2 3m Chapter 6 Skills Practice

13 Lesson 6.2 Skills Practice page 3 Name Date x 2 3x2 3x 20 2 x 18. 2y 2 (27y 2 9z 1 z ) 227y 3 1 9y 2 z 2 y 2 z y (0.3m 1 0.6n) (0.3m 1 0.6n) 0.4(0.3m) 1 0.4(0.6n) m n m n m 1 0.2n (22 1 a 1 b ) ( 22 1 a 1 b ) ( a ) b ( ) ( a ) b ( 33 4 ) a b a b Evaluate each expression for the given value. Choose whether to simplify before evaluating or not. Show your work (24a 1 (213)) for a x 1 3x 2 9 for x (24a 1 (213)) 5 5(24a) 1 5(213) 8x 1 3x (22) 1 3(22) a 1 (265) (26) (2.7) Chapter 6 Skills Practice 567

14 Lesson 6.2 Skills Practice page x(2x 2 14) for x x(2x 2 14) 5 23 ( 7 6 ) ( 2 ( 7 6 ) 2 14 ) ( ) ( 7 3 ) 2 ( ) (14) (4a 1 3.8) for a (4a 1 3.8) 5 5.2(4a) 1 5.2(3.8) (4a) 1 4(3.8) 5 16a (4.3) x for x x 3.2x x 0.4x x 0.4x x a ( a ) for a a ( a ) ( ) ( ( ) ( ) ( ) ) 568 Chapter 6 Skills Practice

15 Lesson 6.3 Skills Practice Name Date Reverse Distribution Factoring Algebraic Expressions Vocabulary Match each term to the correct example. 1. factoring a a. 6(7) 1 6(3) 5 6(7 1 3) 2. common factor b b. the 6 in 6(7) 1 6(3) 3. coefficient e c. 7x and 2x 4. like terms c d. 7x 1 2x 5 9x 5. combining like terms d e. the 4 in 4x Problem Set Rewrite each expression by factoring out the greatest common factor x (8x) 1 8(3) 5 8(8x 1 3) 2. 25y ( y) 2 5(7) 5 25( y 1 7) z 4(9) 1 4(22z) 5 4(9 2 2z) 4. 54n (2n) 1 27(23) 5 27(2n 2 3) a a 218a(2a) 2 18a(21) 5 218a(2a 2 1) 6. 7b ab 7b(b) 1 7b(23a) 5 7b(b 2 3a) 7. 42mn 1 27m 3m(14n) 1 3m(9) 5 3m(14n 1 9) x x 2 2 3x 2x(17x 2 ) 2 x(212x) 2 x(3) 5 2x(17x x 1 3) Chapter 6 Skills Practice 569

16 Lesson 6.3 Skills Practice page c cd 1 45c 15c(2c) 1 15c(2d) 1 15c(3) 5 15c(2c 2 d 1 3) a 1 14b 14(2a) 1 14(b) 5 14(2a 1 b) Simplify each expression by combining like terms. If the expression is already simplified, state how you know x 1 4x 10x y 1 2y 23y m 2 8m 211m 14. 4a 1 8b The expression is already simplified. The terms do not contain the same variables; they are not like terms qr 1 7qr 6qr 16. 9m 2 7m m a 2 1 5a The expression is already simplified. The terms do not contain the same variables; they are not like terms x 2 y 1 4x x The expression is already simplified. The terms do not contain the same variables; they are not like terms s 2 2 5s 2 1 s 2 6s 22s 2 2 5s m m 2 24m Chapter 6 Skills Practice

17 Lesson 6.3 Skills Practice page 3 Name Date Evaluate each expression for the given value. Choose whether or not to factor before evaluating x 1 18 for x ( 1 3 ) x for x (0.5) x 1 60 for x (24) ( x ) for x ( ) ( 4 ) x x for x x(2x 2 5) 5 4 ( ) ( 2 ( ) 2 5 ) ( 212 ) x 1 63x 3 for x x(5 1 9x 2 ) 5 7 ( 2 3 ) ( ( 2 3 ) 2 ) ( ( 4 9 ) ) (9) 5 42 Chapter 6 Skills Practice 571

18 Lesson 6.3 Skills Practice page 4 Evaluate each expression for the given value. Choose whether or not to combine like terms before evaluating. 27. (2y 2 1) 1 (4y 1 8) for y ( 2 ( 1 2 ) 2 1 ) 1 ( 4 ( 1 2 ) 1 8 ) 5 ( ) 1 ( ) (5 1 4y) 1 (3 2 3y) for y y 1 (23y) y (22) x 2 12x 1 4 for x x ( ) x (23.5x) 1 7 for x x 1 (23.5x) x ( x ) 1 ( 1 4 ( (28) ) 1 ( 1 4 x ) for x 5 28 (28) ) 5 ( ) 1 ( ) 572 Chapter 6 Skills Practice

19 Lesson 6.3 Skills Practice page 5 Name Date 32. ( y) 1 (13.3y 2 4.1) for x (24.1) 1 (28.2y) y y (3) Chapter 6 Skills Practice 573

20 574 Chapter 6 Skills Practice

21 Lesson 6.4 Skills Practice Name Date Are They the Same or Different? Verifying That Expressions Are Equivalent Problem Set Determine whether the two expressions in each may be equivalent by evaluating for the given value (9x 1 12) 5 1 (8x 1 4) 1 x for x (9x 1 12) 5 1 (9? ) ( ) (39) (8x 1 4) 1 x 5 1 (8? 3 1 4) (24 1 4) (28) ± 10. The expressions are not equivalent. Chapter 6 Skills Practice 575

22 Lesson 6.4 Skills Practice page (6x 1 0.8) x for x (6x 1 0.8) (6? ) (3.8) x ? The expressions may be equivalent ( 2 x ) 29 ( 2 x x for x ) 5 29 ( 2 5 (25) ) 5 29 ( ) 5 29 ( ) x (25) ± 222. The expressions may be equivalent. 576 Chapter 6 Skills Practice

23 Lesson 6.4 Skills Practice page 3 Name Date 4. 23(1.8x ) 1 1.2x 5 9.3x for x (1.8x ) 1 1.2x 5 23(1.8? ) 1 1.2? ( ) (27.2) x ? ± The expressions may be equivalent ( 26x ) 5 28 ( x ) 1 15x for x ( 26x ) ( 26 ( 1 3 ) ) ( ) ( 14 9 ) ( 1 1 x ) 1 15x 5 28 ( 3 2 ( 1 3 ) ) 1 15 ( 1 3 ) 5 28 ( ) ( ) The expressions may be equivalent. Chapter 6 Skills Practice 577

24 Lesson 6.4 Skills Practice page (3.3x 2 7.1) 1 2.2(1.6x 1 1.8) 5 2(2.5x 1 7.5) 1 0.8(10x ) for x (3.3x 2 7.1) 1 2.2(1.6x 1 1.8) 5 5.4(3.3? ) 1 2.2(1.6? ) 5 5.4( ) 1 2.2( ) 5 5.4(16) 1 2.2(13) (2.5x 1 7.5) 1 0.8(10x ) 5 2(2.5? ) 1 0.8(10? ) 5 2( ) 1 0.8( ) 5 2(25) 1 0.8(81.25) The expressions may be equivalent. 578 Chapter 6 Skills Practice

25 Lesson 6.4 Skills Practice page 5 Name Date Determine whether the two expressions in each are equivalent by simplifying ( x ) 5 2 ( x ) 25 ( x ) 5 2 5(5) 1 (25) ( 4 5 x ) x 2 ( x ) 5 2 ( ) 1 2(22x) x x x. The expressions are equivalent. 8. (6.3x 2 1.4) 1 (3.7x 2 1.6) 5 4(2.5x ) (6.3x 2 1.4) 1 (3.7x 2 1.6) 5 6.3x 1 3.7x 1 (21.4) 1 (21.6) 5 10x 2 3 4(2.5x ) 5 4(2.5x) 1 4(20.75) 5 10x x x 23. The expressions are equivalent. Chapter 6 Skills Practice 579

26 Lesson 6.4 Skills Practice page (23x 1 5) 1 0.6(9x 27.5) 5 18(1.5x 20.2) (23x 1 5) 1 0.6(9x 27.5) 5 0.9(23x) 1 0.9(5) 1 0.6(9x) 1 0.6(27.5) x x x 18(1.5x 2 0.2) (1.5x) 1 18(20.2) x x 2.7x ± 27x. The expressions are not equivalent (2x 2 40) 5 2 ( 1 10 x 2 4 ) 2 2 (2x 2 40) (2x) 1 ( ) (240) x ( 1 10 x 2 4 ) 5 2 ( 1 10 x ) 1 2(24) x x 1 8 ± 1 x 28. The expressions are not equivalent Chapter 6 Skills Practice

27 Lesson 6.4 Skills Practice page 7 Name Date ( x) (5.4x 1 1) 15( x) (20.2) 1 15(22x) (230x) x (5.4x 1 1) (25.5)(5.4x) 1 (25.5)(1) x x x ± 229.7x The expressions are not equivalent ( 6 x ) x ( 1 5 x 2 9) x 1 3 ( 6 10 x ) x ( 6 10 x ) 1 1 3( ) 1 5 x x x ( 1 10 x 29 ) x x 5 2 ( 1 10 x ) 1 2(29) x x 1 5 x x x x 22. The expressions are equivalent. 10 Chapter 6 Skills Practice 581

28 Lesson 6.4 Skills Practice page 8 Determine whether the two expressions in each are equivalent by using graphing technology (7x 2 3) x (7x 3) x The expressions are not equivalent (5x 1 10) (0.25x 1 1) 1 0.5x (5x + 10) - 1 and 4 6(0.25x + 1) + 0.5x The expressions are equivalent. 582 Chapter 6 Skills Practice

29 Lesson 6.4 Skills Practice page 9 Name Date ( 2 x ) 1 2 ( 1 x ) 5 12 ( 4 9 x ) x x 5 ( ) ( 10 x + 5 ) ( ) = x x The expressions are equivalent (1.8x 2 2.6) 1 7x 5 22(2x 1 4) 1 30(0.2x 1 0.3) (1.8x - 2.6) + 7x 2-2(2x + 4) + 30(0.2x + 0.3) The expressions are not equivalent. Chapter 6 Skills Practice 583

30 Lesson 6.4 Skills Practice page ( x 1 10 ) ( x 1 14 ) (1 5 3 x + 10 ) = ( x + 14) The expressions are not equivalent x 1 4(x 2 2) ( x ) (4x 2 2) x + 4 (x 2) = ( x 4 ) +1 ( ) 2 2 4x The expressions are equivalent. 584 Chapter 6 Skills Practice

31 Lesson 6.5 Skills Practice Name Date It Is Time to Justify! Simplifying Algebraic Expressions Using Operations and Their Properties Problem Set Complete each table. Either simplify the left side using the given operation or property, or use an operation or a property to justify each simplification step. Indicate if the expressions are equivalent (22x 1 5) 1 2(0.5x 1 2) 5 9x 2 16 Step Justification 24(22x 1 5) 1 2(0.5x 1 2) 5 Given 8x 1 (220) 1 x x 1 x 1 (220) x 2 16 Distributive Property of Multiplication over Addition Commutative Property of Addition Addition of Like Terms Yes, they are equivalent. Chapter 6 Skills Practice 585

32 Lesson 6.5 Skills Practice page x x x Step Justification 5x x Given 6 3 5x x Distributive Property of Division over Subtraction 5 x x 1 ( ) 1 ( ) 5 Commutative Property of Addition 17 6 x Addition of Like Terms Yes, they are equivalent (3.2x 2 5.3) (5.8x 2 3.6) 5 86x 2 67 Step 4.1(3.2x 2 5.3) (5.8x 2 3.6) x x Justification Given Distributive Property of Multiplication over Subtraction 13.12x x 1 (221.73) 1 (245.72) 5 Commutative Property of Addition 86.78x Addition of Like Terms No, they are not equivalent. 586 Chapter 6 Skills Practice

33 Lesson 6.5 Skills Practice page 3 Name Date 4. 7(2 2 3x) 2 5(6 1 x) 1 4x x Step Justification 7(2 2 3x) 2 5(6 1 x) 1 4x 5 Given x x 1 4x 5 Distributive Property of Multiplication over Addition and Subtraction 14 1 (230) 1 (221x) 1 (25x) 1 4x 5 Commutative Property of Addition x Addition of Like Terms Yes, they are equivalent x x 4 5 x Step 12x x Justification Given 12x Distributive Property of x Division over Addition and Subtraction 3x x 5 Divide 4 3x 2 3x ( ) 5 Commutative Property of Addition Addition of Like Terms No, they are not equivalent. Chapter 6 Skills Practice 587

34 Lesson 6.5 Skills Practice page [7x 1 2(5 1 x)] 5 224x 2 30 Step Justification 23[7x 1 2(5 1 x)] 5 Given 23[7x x] 5 23[7x 1 2x 1 10] 5 Distributive Property of Multiplication over Addition Commutative Property of Addition 23[9x 1 10] 5 Addition of Like Terms 227x 2 30 Distributive Property of Multiplication over Addition No, they are not equivalent. 588 Chapter 6 Skills Practice

35 Lesson 6.5 Skills Practice page 5 Name Date 7. 8x 1 3(7 1 x) 1 (9x 2 1) 5 2x Step 8x 1 3(7 1 x) 1 (9x 2 1) x x 1 9x 1 (21) x 1 3x x 1 (21) x x 1 (21) Justification Given Distributive Property of Multiplication over Addition Commutative Property of Addition Addition of Like Terms 11x x 1 (21) Distribution of Division over Addition and Subtraction 11 x x ( ) 5 Commutative Property of Addition x Addition of Like Terms 10 2x 1 2 Division Yes, they are equivalent. Chapter 6 Skills Practice 589

36 Lesson 6.5 Skills Practice page ( x) 1 0.5[ ( x) x Step Justification 6( x) 1 0.5[ ( x) Given x 1 0.5[ x) Distributive Property of Multiplication over Subtraction and Addition x 1 0.5( x) Subtraction x x Distributive Property of Multiplication over Subtraction (23.6) 1 (21.2x) 1 (24.2x) 5 Commutative Property of Addition x Addition of Like Terms Yes, they are equivalent. Simplify the left side and/or right side completely to determine if the two expressions are equivalent. Use an operation or a property to justify each step and indicate if the expressions are equivalent (24x 1 8) (25x 1 7) 5 9x 2 17 Step Justification 26(24x 1 8) (25x 1 7) 5 Given 24x 1 (248) (215x) Distributive Property of Multiplication over Addition 24x 1 (215x) 1 (248) Commutative Property of Addition 9x 2 17 Addition of Like Terms Yes, they are equivalent. 590 Chapter 6 Skills Practice

37 Lesson 6.5 Skills Practice page 7 Name Date 10. 9(0.3x 2 2.2) 1 6(0.3x 2 2.2) 5 3(1.5x 2 11) Step 9(0.3x 2 2.2) 1 6(0.3x 2 2.2) 5 Justification Given 15(0.3x 2 2.2) 5 Factoring Distributive Property of Mult. over Add. 4.5x Distributive Property of Mult. oversubt. 3(1.5x 2 11) Factoring Yes, they are equivalent (6x 1 2) 1 7(12 1 8x) 5 6x Step 5 2 4(6x 1 2) 1 7(12 1 8x) (224x) 1 (28) x (224x) x Justification Given Distributive Property of Multiplication over Addition Combine Like Terms ( 2 24x 3 ) x 4 5 Distributive Property of Division over Addition (28x) x 5 Divide 28x 1 14x 1 (21) Commutative Property of Addition 6x 1 20 Combine Like Terms No, they are not equivalent. Chapter 6 Skills Practice 591

38 Lesson 6.5 Skills Practice page (2x 2 11) 2 5(x 1 9) 1 13x 5 x 2 7(x 1 2) 1 2(9x 1 10) 1 1 Left Side Step Justification 8 2 4(2x 2 11) 2 5(x 1 9) 1 13x 5 Given 8 1 4x x x 5 Distributive Property of Mult. over Add. and Subt. 4x 1 (25x) 1 13x (245) 5 Commutative Property of Addition 12x 1 7 Combine Like Terms Right Side Step x 2 7(x 1 2) 1 2(9x 1 10) 1 1 Justification Given x 2 7x x Distributive Property of Multiplication over Addition x 1 (27x) 1 18x 1 (214) Commutative Property of Addition 12x 1 7 Combine Like Terms Yes, they are equivalent. 592 Chapter 6 Skills Practice

39 Lesson 6.5 Skills Practice page 9 Name Date (23.2x 1 2.4) 1 6[ (0.9x 1 0.8)] 5 3.8(2x 2 7) 1 1.4(6 1 3x) Left Side Step Justification 0.5(23.2x 1 2.4) 1 6[ (0.9x 1 0.8)] 5 Given 21.6x ( x 1 3.2) 5 Distributive Property of Multiplication over Addition 21.6x (3.6x 1 4.9) 5 Combine Like Terms 21.6x x Distributive Property of Multiplication over Addition 21.6x x Commutative Property of Addition 20x Combine Like Terms Right Side Step 3.8(2x 2 7) 1 1.4(6 1 3x) 5 Justification Given 7.6x x 5 Distributive Property of Multiplication over Addition 7.6x 1 4.2x 1 (226.6) Commutative Property of Addition 11.8x Combine Like Terms No, they are not equivalent. Chapter 6 Skills Practice 593

40 Lesson 6.5 Skills Practice page (16x 2 8) x x (29x 2 12) ( x) Left Side Step Justification (16x 2 8) x Given x x Distributive Property of Mult. over Subt (214) x 1 ( x ) 5 Commutative Property of Addition x Combine Like Terms 8 2 Right Side Step Justification x (29x 2 12) ( x) Given x 1 21x x Distributive Property of Mult. 2 8 over Add. and Subt x 1 21x 1 22x (266) Commutative Property of Addition x Combine Like Terms x Commutative Property of Addition 8 2 Yes, they are equivalent. 594 Chapter 6 Skills Practice