Skills Practice - Mrs. Cruz's Algebra Classmrscruzalgebra.weebly.com/uploads/8/9/5/6/8956636/c4_review...368 Chapter 4 l Skills Practice 26. An employee at a clothing store can fold

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Chapter 4 l Skills Practice 361

Skills Practice Skills Practice for Lesson 4.1

Name _____________________________________________ Date _________________________

Up, Up, and Away!Solving and Graphing Inequalities in One Variable

Vocabulary Provide an example of each term.

1. inequality

x � 2

2. inequality symbol

�

3. compound inequality

0 � x � 8

4. graph of an inequality

543–2 –1 –0 1 2

5. solution of an inequality

For the inequality x � 2, 3 is a solution, because the inequality is true when x � 3.

Problem Set

Write an inequality to represent each situation.

1. The temperature in March is greater than or equal to 45 degrees Fahrenheit. Let x represent

the temperature in March.

x � 45

2. May’s textbook weighed greater than 7 pounds. Let x represent the weight of May’s

textbook.

x � 7

3. It took Chase less than 3 hours to complete his homework. Let x represent the amount of

time it took Chase to complete his homework.

x � 3

4. Marissa slept for 8 hours or less. Let x represent the number of hours Marissa slept.

x � 8

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362 Chapter 4 l Skills Practice

Write a compound inequality to represent each situation.

5. To ride a rollercoaster, your height must be greater than or equal to 48 inches, and less

than or equal to 78 inches. Let x represent the range of heights that are permitted on the

rollercoaster.

48 � x � 78

6. To avoid an extra charge for your airline luggage, your bag must weigh less than 50 pounds.

Your bag weighs greater than 0 pounds. Let x represent the weight of your luggage that will

avoid an extra charge.

0 � x � 50

7. The recommended ages for a board game are greater than 4 and less than or equal to 12.

Let x represent the recommended ages for the board game.

4 � x � 12

8. A recipe serves greater than or equal to 2 people, but fewer than 8 people. Let x represent

the number of people the recipe will serve.

2 � x � 8

Write the inequality for each graph shown.

9.

2 3 410–5–6 –4 –3 –2 –1

x � �1

10.

1 2 3 4 5 6 7 8 9 10 11

x � 7

11.

–8 –7 –6 –5 –4 –3 –2 –1 0 1 2

x � �6 or x � �3

12. 9 10 11 12 13 14 15 16 17 18 19

x � 11 or x � 16

13.

3 4 5 6 7 8 9 11 12 1310

7 � x � 12

14.

–13 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3

�11 � x � �5

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Name _____________________________________________ Date _________________________

15.

–7–8 –6 –5 –4 –3 –2 –1 0 1 2

�5 � x � �2

16.

6 7 8 9 10 11 12 13 14 15 16

8 � x � 15

17.

–5 –4 –3 –2 –1 0 1 2 3 4 5

x � �3 or x � 0

18.

–13 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3

x � �11 or x � �9

Solve each inequality and graph the solution.

19. 6x < 12

x � 2

2 3 410–3 –2 –1

20. 12x > 72

x � 6

3 4 5 6 7 8 9 10

21. � x __ 2 ≤ 4

x � �8

–9–10–11–12–13 –8 –7 –6

22. x __ 4 < �1

x � �4

–7 –6 –5 –4 –3 –2 –1 0

23. 5x � 6 < 31

5x � 25

x � 5

32 4 5 6 7 8 9

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24. �3x � 4 ≥ 23

�3x � 27

x � �9

–9–10–11–12–13 –8 –7 –6

25. �2 � x __ 3 ≥ �4

� x __ 3 � �2

x � 6

32 4 5 6 7 8 9

26. 14 � x __ 4 ≤ 17

� x __ 4 � 3

x � �12

–9–10–11–12–13 –8 –7 –6

27. �6x � 12 < 12

�6x � 24

x � �4

–7 –6 –5 –4 –3 –2 –1 0

28. 7x � 18 < 74

7x � 56

x � 8

32 4 5 6 7 8 9

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Name _____________________________________________ Date _________________________

Moving a Sand PileRelations and Functions

Vocabulary Match each definition to its corresponding term.

1. an indication that a group of numbers is part of a set a. dependent variable

i. set notation

2. the variable that represents the input value of a function b. domain

d. independent variable

3. the set of all output values for a function c. function

g. range

4. the variable that represents the output value of a function d. independent variable

a. dependent variable

5. any set of ordered pairs e. input

h. relation

6. a relation in which for every input there is exactly f. output

one output

c. function

7. the set of all input values for a function g. range

b. domain

8. the second coordinate of an ordered pair in a relation h. relation

f. output

9. the first coordinate of an ordered pair in a relation i. set notation

e. input

Problem Set Identify the inputs and outputs of each relation.

1. Relation: (�3, 2), (2, �3), (4, 0), (5, 5)

inputs: �3, 2, 4, 5; outputs: 2, �3, 0, 5

2. Relation: (0, 1), (1, 2), (2, 3), (3, 4)

inputs: 0, 1, 2, 3; outputs: 1, 2, 3, 4

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3. Relation: (�2, �2), (2, �1), (3, 0), (5, 3), (5, 4)

inputs: �2, 2, 3, 5; outputs: �2, �1, 0, 3, 4

4. Relation: (�3, �1), (�2, 0), (�1, 3), (�1, 0), (0, �1)

inputs: �3, �2, �1, 0; outputs: �1, 0, 3

5. Relation: (4, 11), (5, 18), (6, 25), (7, 32), (8, 39)

inputs: 4, 5, 6, 7, 8; outputs: 11, 18, 25, 32, 39

6. Relation: (10, 1), (11, 6), (12, 11), (13, 16), (14, 21)

inputs: 10, 11, 12, 13, 14; outputs: (1, 6, 11, 16, 21)

7. Relation: (�4, �2), (�4, �6), (�7, �9), (�7, �11), (�8, �15)

inputs: �4, �7, �8; outputs: �2, �6, �9, �11, �15

8. Relation: (1, �14), (2, �19), (2, �24), (3, �26), (3, �31)

inputs: 1, 2, 3; outputs: �14, �19, �24, �26, �31

Determine if each relation is a function.

9. Relation: (�2, 2), (3, 2), (4, �1), (5, �2)

This relation is a function because each input has only one output.

10. Relation: (3, 0), (4, 3), (4, 4), (6, 7)

This relation is not a function because the input 4 has two outputs, 3 and 4.

11. Relation: (3, 3), (4, 1), (4, 6), (5, 0), (7, 5)

This relation is not a function because the input 4 has two outputs, 1 and 6.

12. Relation: (�1, 2), (2, 3), (3, 0), (4, 1), (5, 0)


13. Relation: (�4, �7), (�4, �4), (�2, �1), (�1, 2), (5, �10)

This relation is not a function because the input �4 has two outputs, �7 and �4.

14. Relation: (1, 8), (1, �8), (2, 11), (3, 5), (9, 6)

This relation is not a function because the input 1 has two outputs, 8 and �8.

15. Relation: (�1, 4), (4, 10), (14, 10), (20, 14), (21, 20)


16. Relation: (�5, 0), (�4, �3), (�3, �5), (�2, �7), (�1, 0)


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Name _____________________________________________ Date _________________________

If the relation is a function, identify the domain and range. If the relation is not a function, explain why.

17. Relation: (�2, 5), (0, 7), (2, 9), (4, 11), (6, 13)

The relation is a function.

The domain is {�2, 0, 2, 4, 6}. The range is {5, 7, 9, 11, 13}.

18. Relation: (�5, 0), (�4, �1), (�3, �2), (�2, �3), (�1, �4)


The domain is {�5, �4, �3, �2, �1}. The range is {0, �1, �2, �3, �4}.

19. Relation: (0, 10), (0, 11), (1, 14), (2, 16), (3, 18)

The relation is not a function because the input value 0 has two outputs, 10 and 11.

20. Relation: (13, 2), (15, 8), (17, 14), (19, 20), (21, 26)


The domain is {13, 15, 17, 19, 21}. The range is {2, 8, 14, 20, 26}.

21. Relation: (1, 6), (2, �5), (3, �12), (4, �19 ), (5, �26)


The domain is {1, 2, 3, 4, 5}. The range is {6, �5, �12, �19, �26}.

22. Relation: (6, 9), (9, 11), (9, 14), (13, 28), (19, 30)

The relation is not a function because the input value 9 has two outputs, 11 and 14.

23. Relation: (11, 8), (12, 7), (13, 6), (14, 6), (15, 4)


The domain is {11, 12, 13, 14, 15}. The range is {8, 7, 6, 4}.

24. Relation: (�5, 0), (�3, 8), (�1, 8), (0, 5), (1, 6)


The domain is {�5, �3, �1, 0, 1}. The range is {0, 8, 5, 6}.

Identify the independent and dependent variables in each situation.

25. The cost in dollars, c, to rent an inflatable bouncer for any number of days, d, is represented

by the equation c � 85 � 30d.

The independent variable is the number of days the inflatable bouncer is rented. The dependent variable is the total cost in dollars.

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26. An employee at a clothing store can fold n shirts in t minutes, which is represented by the

equation t � 3n � 6.

The independent variable is the number of shirts folded. The dependent variable is the total time in minutes to fold the shirts.

27. A sketch artist can draw s sketches in t minutes, which is represented by the equation

t � 15s � 30.

The independent variable is the number of sketches drawn. The dependent variable is the total time in minutes to draw the sketches.

28. The total cost, c, in dollars that Zoe charges to wrap p presents is represented by the

equation c � 8 � 2p.

The independent variable is the number of presents Zoe wraps. The dependent variable is the total cost in dollars.

29. The number of oranges remaining, r, can be represented by the equation r � 30 � n, where

n is the number of oranges eaten.

The independent variable is the number of oranges eaten. The dependent variable is the number of oranges remaining.

30. The number of empty theater seats, s, can be determined by the equation s � 2000 � t, where t is the number of tickets purchased.

The independent variable is the number of tickets purchased. The dependent variable is the number of seats remaining.

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Name _____________________________________________ Date _________________________

Let’s Bowl!Evaluating Functions, Function Notation, Domain, and Range

VocabularyWrite the term from the box that best completes each statement.

domain evaluate a function function function notation range

1. To evaluate a function is to replace the variable with the given value and calculate

the result.

2. A(n) function is a relation in which every input has exactly one output.

3. A method of writing functions such that the dependent variable is replaced with the name of

the function is called function notation .

4. The range is the set of all output values of a function.

5. The domain is the set of all input values of a function.

Problem Set Use function notation to write an equation for each situation.

1. You decide to hit balls at a golf range. You rent a club for $5 and the cost of each ball is

$0.10. Write a function f that represents the total cost in dollars to hit x golf balls.

f(x) � 5 � 0.10x

2. You and your friends go ice skating. Because you don’t own ice skates, you need to rent

them for $7.50. Each hour of skating costs $2.50. Write a function f that represents your

total cost in dollars if you skate for x hours.

f(x) � 7.50 � 2.50x

3. Dominic gives Blake a 5-meter head start in a race. Blake runs 11 meters in one second.

Write a function f that represents Blake’s position in meters in the race after x seconds.

f(x) � 5 � 11x

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4. To do household repairs, a carpenter charges a fixed fee of $30 plus $15 for each hour that

he works. Write a function f that represents the total charge in dollars for x hours of work.

f(x) � 30 � 15x

Evaluate each function at the specified value.

5. f(x) � x � 13 at x � �1 6. f(x) � 6x at x � 12

f(�1) � �1 � 13 f(12) � 6(12)

� 12 � 72

7. f(x) � 18 � 4x at x � 9 8. f(x) � x __ 7 � 5 at x � 21

f(9) � 18 � 4(9) f(21) � 21 ___ 7 � 5

� 18 � 36 � 3 � 5

� �18 � 8

9. f(x) � 25 � x __ 3 at x � 30 10. f(x) � 11x � 8 at x � �10

f(30) � 25 � 30 ___ 3 f(10) � 11(�10) � 8

� 25 � 10 � �110 � 8

� 15 � �118

11. f(x) � �x � 15 at x � �1 12. f(x) � � x __ 8 � 9 at x � �40

f(�1) � �(�1) � 15 f(40) � � (�40)

______ 8 � 9

� 1 � 15 � 5 � 9

� �14 � �4

Determine the corresponding range of each function for the given domain.

13. f(x) � 5x 14. f(x) � 10 � x

Domain: {3, 4, 5, 6} Domain: {9, 10, 11, 12}

f(3) � 5(3) � 15 f(9) � 10 � 9 � 19

f(4) � 5(4) � 20 f(10) � 10 � 10 � 20

f(5) � 5(5) � 25 f(11) � 10 � 11 � 21

f(6) � 5(6) � 30 f(12) � 10 � 12 � 22

Range: {15, 20, 25, 30} Range: {19, 20, 21, 22}

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Name _____________________________________________ Date _________________________

15. f(x) � x __ 4 � 6 16. f(x) � 12x � 2

Domain: {2, 4, 6, 8} Domain: 1 __ 2 , 1, 3 __

2 , 2

f(2) � 2 __ 4 � 6 � 1 __

2 � 6 � � 11 ___

2 f ( 1 __

2 ) � 12 ( 1 __

2 ) � 2 � 6 � 2 � 8

f(4) � 4 __ 4 � 6 � 1 � 6 � �5 f(1) � 12(1) � 2 � 12 � 2 � 14

f(6) � 6 __ 4 � 6 � 3 __

2 � 6 � � 9 __

2 f ( 3 __

2 ) � 12 ( 3 __

2 ) � 2 � 18 � 2 � 20

f(8) � 8 __ 4 � 6 � 2 � 6 � �4 f(2) � 12(2) � 2 � 24 � 2 � 26

Range: � � 11 ___ 2 , �5, � 9 __

2 , �4 � Range: {8, 14, 20, 26}

17. f(x) � 3 � 5x 18. f(x) � �x � 20

Domain: {any real number} Domain: {any real number}

Range: {any real number} Range: {any real number}

19. f(x) � �3x � 1 20. f(x) � � x __ 5 � 2

Domain: 4, 25 ___ 6 , 5, 46 ___

9 Domain: {�1, 0, 1, 2}

f(4) � �3(4) � 1 � �12 � 1 � �11 f(�1) � � (�1)

_____ 5

� 2 � 1 __ 5 � 2 � 11 ___

5

f ( 25 ___ 6 ) � �3 ( 25 ___

6 ) � 1 � � 25 ___

2 � 1 � � 23 ___

2 f(0) � 0 __

5 � 2 � 0 � 2 � 2

f(5) � �3(5) � 1 � �15 � 1 � �14 f(1) � � 1 __ 5 � 2 � 9 __

5

f ( 46 ___ 9 ) � �3 ( 46 ___

9 ) � 1 � � 46 ___

3 � 1 � � 43 ___

3 f(2) � � 2 __

5 � 2 � 8 __

5

Range: � �11, � 23 ___ 2 , �14, � 43 ___

3 � Range: � 11 ___

5 , 2, 9 __

5 , 8 __

5 �

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Use function notation to write an equation for each situation. Then use the equation to answer the question.

21. A moving company charges a fixed fee of $100 for a moving truck plus $35 for each mover

that is hired. Write a function f that represents the total cost in dollars if a person hires

x movers. What is the total cost if a person hires 4 movers?

f(x) � 100 � 35x

f(4) � 100 � 35(4) � 100 � 140 � 240

If a person hires 4 movers, the total cost is $240.

22. Your little brother is having a party at the local zoo. The zoo charges a party fee of $50 plus

$5 for each guest. Write a function f that represents the total cost in dollars for x guests.

What is the total cost for 13 guests?

f(x) � 50 � 5x

f(13) � 50 � 5(13) � 50 � 65 � 115

The total cost for 13 guests is $115.

23. It takes Mrs. Won 20 minutes to get her desk ready to grade homework. It takes her

10 minutes to grade each page of homework. Write a function f that represents the total

time, in minutes, that Mrs. Won takes to grade x pages of homework. How long does it take

her to grade 25 pages of homework?

f(x) � 20 � 10x

f(25) � 20 � 10(25) � 20 � 250 � 270

It takes Mrs. Won 270 minutes to grade 25 pages of homework.

24. It takes Eric 3 minutes to find each item he needs at the grocery store, and a set time of

12 minutes to purchase all his items. Write a function f that represents the total time, in

minutes, Eric will spend at the grocery store if he needs x items. How long will Eric be at the

grocery store if he needs 30 items?

f(x) � 12 � 3x

f(30) � 12 � 3(30) � 12 � 90 � 102

Eric will spend 102 minutes, or 1 hour 42 minutes, at the grocery store if he needs 30 items.

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Name _____________________________________________ Date _________________________

Math MagicThe Distributive Property

Vocabulary Explain the similarities and differences between each set of terms.

1. common factor and greatest common factor

A common factor and a greatest common factor are both whole numbers that are factors of two or more integers or expressions. The greatest common factor is the largest common factor between two or more integers or expressions. All greatest common factors are common factors, but not all common factors are greatest common factors.

2. combine like terms and simplify

To combine like terms means to add or subtract like terms in an expression or equation. When simplifying an expression, you rewrite the expression as an equivalent yet briefer expression, using addition, subtraction, multiplication, or division. Combining like terms is one way to simplify an expression or equation.

3. terms and like terms

A term is any member of a sequence. Like terms are also members of a sequence, but their variable portions must be the same.

4. distributive property and factoring

The distributive property states that a(b � c) � ab � ac. Factoring uses the distributive property in reverse. A factored expression is equivalent to its original expression because you can use the distributive property to obtain the original expression.

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Problem Set Use the distributive property to simplify each expression.

1. 3(9 � 14) 2. 8(12 � 7)

3(9) � 3(14) � 27 � 42 � 69 8(12) � 8(7) � 96 � 56 � 40

3. 24 � 32 ________ 4 4. 18 � 72 ________

6

24 ___ 4 � 32 ___

4 � 6 � 8 � �2 18 ___

6 � 72 ___

6 � 3 � 12 � 15

5. 11(2x � 5) 6. 7(3x � 12)

11(2x) � 11(5) � 22x � 55 7(3x) � 7(12) � 21x � 84

7. �14 � 6x _________ 2 8. 35 � 28x _________

7

� 14 ___ 2 � 6x ___

2 � �7 � 3x 35 ___

7 � 28x ____

7 � 5 � 4x

Use the distributive property to factor the greatest common factor from each algebraic expression.

9. 26x � 39 10. 18 � 27x

26x � 39 � 13(2x) � 13(3) 18 � 27x � 9(2) � 9(3x)

� 13(2x � 3) � 9(2 � 3x)

11. 100x � 80 12. 60x � 12

100x � 80 � 20(5x) � 20(4) 60x � 12 � 12(5x) � 12(1)

� 20(5x � 4) � 12(5x � 1)

13. 40 � 16x 14. 26 � 6x

40 � 16x � 8(5) � 8(2x) 26 � 6x � 2(13) � 2(3x)

� 8(5 � 2x) � 2(13 � 3x)

15. 35x � 55 16. 49x � 28

35x � 55 � 5(7x) � 5(11) 49x � 28 � 7(7x) �7(4)

� 5(7x � 11) � 7(7x � 4)

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Name _____________________________________________ Date _________________________

Use the distributive property to simplify each algebraic expression.

17. 33x � 55x 18. 98x � 28x

33x � 55x � 11(3x � 5x) 98x � 28x � 14(7x � 2x)

� 11(�2x) � 14(5x)

� �22x � 70x

19. 6x � 36x 20. 80x � 2x

6x � 36x � 6(x � 6x) 80x � 2x � 2(40x � x)

� 6(7x) � 2(41x)

� 42x � 82x

21. 27x � 81x 22. 25x � 40x

27x � 81x � 27(x � 3x) 25x � 40x � 5(5x � 8x)

� 27(�2x) � 5(�3x)

� �54x � �15x

23. 42x � 26x 24. 68x � 60x

42x � 26x � 2(21x � 13x) 68x � 60x � 4(17x � 15x)

� 2(8x) � 4(2x)

� 16x � 8x

Write two expressions for the total area of the two rectangles. Then calculate the total area.

25.

10 ft 5 ft

6 ft

6(15) � 90; area is 90 square feet.

6(10) � 6(5) � 60 � 30 � 90; area is 90 square feet.

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26.

20 m 4 m

8 m

8(24) � 192; area is 192 square meters.

8(20) � 8(4) � 160 � 32 � 192; area is 192 square meters.

27.

20 cm 2 cm

3 cm

3(22) � 66; area is 66 square centimeters.

3(20) � 3(2) � 60 � 6 � 66; area is 66 square centimeters.

28.

10 in. 8 in.

8 in.

8(18) � 144; area is 144 square inches.

8(10) � 8(8) � 80 � 64 � 144; area is 144 square inches.

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Name _____________________________________________ Date _________________________

Numbers in Your Everyday LifeReal Numbers and Their Properties

Vocabulary Give an example of each term.

1. rational number 2. repeating decimal

1 __ 2 0.333333... � 0.

__ 3

3. irrational number 4. real number

√__

2 �1

5. Venn diagram 6. a property of the real number system

Wholenumbers

Integers closure

Venn diagram showing the relationship between integers and whole numbers.

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7. whole number 8. natural number

0 2

9. integer 10. closure in the real number system under

multiplication

�3 4 · 1 __ 2 � 2

Two real numbers multiplied together result in a real number.

11. additive identity in the real number system 12. multiplicative identity in the real number

system

4 � 0 � 4 8 · 1 � 8

The additive identity of 4 is 0. The multiplicative identity of 8 is 1.

13. additive inverse in the real number system 14. multiplicative inverse in the real number

system

3 � (�3) � 0 2 · 1 __ 2 � 1

The additive inverse of 3 is �3. The multiplicative inverse of 2 is 1 __ 2 .

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Name _____________________________________________ Date _________________________

Problem Set Write each repeating decimal as a rational number.

1. 0.6666… 2. 0.0202…

10w � 6.6666… 100w � 2.0202…

�w � 0.6666… �w � 0.0202…

9w � 6 99w � 2

w � 6 __ 9 � 2 __

3 w � 2 ___

99

3. 0.1313… 4. 0.5555…

100w � 13.1313… 10w � 5.5555…

�w � 0.1313… �w � 0.5555…

99w � 13 9w � 5

w � 13 ___ 99

w � 5 __ 9

5. 0.1111… 6. 0.2727…

10w � 1.1111… 100w � 27.2727…

�w � 0.1111… �w � 0.2727…

9w � 1 99w � 27

w � 1 __ 9 w � 27 ___

99 � 3 ___

11

7. 0.0505… 8. 0.4545…

100w � 5.0505… 100w � 45.4545…

�w � 0.0505… �w � 0.4545…

99w � 5 99w � 45

w � 5 ___ 99

w � 45 ___ 99

� 5 ___ 11

Graph the numbers on a number line. Then list the numbers from least to greatest.

9. 0, � 4 __ 3 , 2.

__

2 , �0.5 10. 12 ___ 11

, �1. ___

98 , 3, 0.456732...

–2 –1 0 1 2

–3 –2 –1 0 1 2 3

� 4 __ 3 , �0.5, 0, 2.

__ 2 �1.

___ 98 , 0.456732..., 12 ___

11 , 3

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11. � 1 __

3 , 2.

___

56 , 6 __ 7 , 0.894375... 12. 1.

___

47 , �0.7, 13 ___ 12

, �

–3 –2 –1 0 1 2 3

–3 –2 –1 0 1 2 3

� 1 __

3 , 6 __

7 , 0.894375..., 2.

___ 56 �0.7, 13 ___

12 , 1.

___ 47 ,

Classify each real number as irrational, rational, integer, whole number, or natural number. Use all relevant terms.

13. 4.392573… 14. �1. __

3

irrational rational

15. 1 ____ 125

16. �7

rational rational and integer

17. 15 18. 9.325407…

rational, integer, whole number, irrationaland natural number

19. 0 20. 45 ___ 46

rational, integer, whole number rational

21. �550 22. 310

rational and integer rational, integer, whole number, and natural number

Answer each question about real numbers. If your answer is no, give an example that supports your reasoning.

23. Is every natural number an integer?

Yes. Natural numbers are a certain type of integer.

24. Is every real number a rational number?

No. There are two types of real numbers: rational and irrational. For example, is a real number that is not rational.

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Name _____________________________________________ Date _________________________

25. Is every integer a whole number?

No. There are negative integers, and whole numbers are either 0 or positive. For example, �3 is an integer that is not a whole number.

26. Is every integer a real number?

Yes. Integers are a certain type of real number.

For each equation, identify the property that is used in each step.

27. 2(x � 15) � � 138 ____ 3 Given problem

2x � 30 � � 138 ____ 3 Distributive Property of Multiplication Over Subtraction

2x � 30 � �46 Divide.

2x � 30 � 30 � �46 � 30 Add 30 to each side.

2x � �16 Add.

2x ___ 2 � � 16 ___

2 Divide each side by 2.

x � �8 Divide.

28. 10 � (x � 4) � 3 � �5 Given problem

10 � x � (4 � 3) � �5 Associative Property of Addition

10 � x � 7 � �5 Add.

x � (10 � 7) � �5 Commutative and Associative Property of Addition

x � 17 � �5 Add.

x � 17 � 17 � �5 � 17 Subtract 17 from each side.

x � �22 Subtract.

29. 5(3 � x) � 1 � 71 Given problem

15 � 5x � 1 � 71 Distributive Property of Multiplication Over Addition

5x � (15 � 1) � 71 Commutative and Associative Property of Addition

5x � 16 � 71 Add.

5x � 16 � 16 � 71 � 16 Subtract 16 from each side.

5x � 55 Subtract.

5x ___ 5 � 55 ___


x � 11 Divide.

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30.

4(x � 5) ________

2 � 3 � 5 Given problem

4(x � 5)

________ 2 � �2 Subtract.

4x � 20 ________ 2 � �2 Distributive Property of Multiplication Over

Subtraction

4x ___ 2 � 20 ___

2 � �2 Distributive Property of Division Over Subtraction

2x � 10 � �2 Divide.

2x � 10 � 10 � �2 � 10 Add 10 to each side.

2x � 8 Add.

2x ___ 2 � 8 __


x � 4 Divide.

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Name _____________________________________________ Date _________________________

Technology ReporterSolving More Complicated Equations

Vocabulary Write a definition for each term in your own words.

1. solve an equation

To solve an equation, isolate a variable in an equation to determine what value of the variable will make the equation true.

2. simplify an expression

To simplify an expression, rewrite the expression as an equivalent yet briefer expression that is easier to work with.

Problem Set Simplify each side of the equation. Do not solve.

1. 5(3x � 12) � 3x 2. 18(x � 2) � 5x

15x � 60 � 3x 18x � 36 � 5x

3. 14(8 � 9x) � 3(1 � 10x) 4. 12(4 � 7x) � 2(6 � 11)

112 � 126x � 3 � 30x 48 � 84x � 12 � 22

48 � 84x � � 10

5. 25x � 60 _________ 5 � 4(13 � 2x) � 2 6. 7(9 � 15x) � 1 � 40x � 24 _________

8

25x ____ 5 � 60 ___

5 � 52 � 8x � 2 63 � 105x � 1 � 40x ____

8 � 24 ___

8

5x � 12 � 50 � 8x 105x � 64 � 5x � 3

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Determine if each given value of x is a solution to the equation. If it is not, solve for x.

7. 17(x � 2) � 30 � x; x � 4 8. 2(14x � 21)

___________ 7 � x � 6; x � 2

17(4 � 2) � 30 � 4 2(14(2) � 21)

____________ 7 � 2 � 6

68 � 34 � 34 2(28 � 21)

__________ 7 � �4

34 � 34 56 ___ 7 � 42 ___

7 � �4

x � 4 is a solution of the equation. 8 � 6 � �4

14 � �4

x � 2 is not a solution of the equation.

2(14x � 21)

___________ 7 � x � 6

28x ____ 7 � 42 ___

7 � x � 6

4x � 6 � x � 6

3x � 6 � �6

3x � �12

x � �4

9. 16 � x � 2(20 � x); x � 5 10. 2(x � 7) � 4(9x � 3)

_________ 3 ; x � 1

16 � 5 � 2(20 � 5) 2(1 � 7) � 4(9(1) � 3)

__________ 3

21 � 40 � 10 2 � 2(7) � 4(9)

____ 3 �

4(3) ____

3

21 � 30 2 � 14 � 36 ___ 3 � 12 ___

3

x � 5 is not a solution of the equation. 16 � 12 � 4

16 � x � 2(20 � x) 16 � 16

16 � x � 40 � 2x x � 1 is a solution of the equation.

16 � 3x � 40

3x � 24

x � 8

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Name _____________________________________________ Date _________________________

Solve each equation.

11. 2x � 8 � 14 � x 12. 5 � 3x � 20 � 2x

3x � 8 � 14 5 � 20 � 5x

3x � 6 �15 � 5x

x � 2 �3 � x

13. 4(x � 1) � 2x 14. �x � �3(x � 4)

4x � 4 � 2x �x � �3x � 12

�4 � �2x 2x � �12

2 � x x � �6

15. 8 � 9x � 3(1 � 10x) 16. 12(4 � 7x) � 6 � 11x

8 � 9x � 3 � 30x 48 � 84x � 6 � 11x

8 � 3 � 39x 48 � 95x � 6

5 � 39x 95x � �42

5 ___ 39

� x x � � 42 ___ 95

17. 13(3x � 2) � 2 � 14x � 1 18. 5 � 2x � 10(2 � 3x) � 8

39x � 26 � 2 � 14x � 1 5 � 2x � 20 � 30x � 8

39x � 24 � 14x � 1 5 � 2x � 12 � 30x

25x � 24 � 1 5 � 28x � 12

25x � 25 28x � 7

x � 1 x � 1 __ 4 � 0.25

19. �2(6x � 1) � �3(3 � x) 20. �(9 � 11x) � 5(2x � 20)

�12x � 2 � �9 � 3x �9 � 11x � 10x � 100

�2 � �9 � 15x �9 � 21x � 100

7 � 15x 91 � 21x

7 ___ 15

� x 13 ___ 3 � x

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Write an equation to represent each situation. Then use the equation to answer the question.

21. Cody is trying to decide where to take a trip. He wants to compare his total flight and hotel

costs between two destinations. If he travels to San Diego, his flight will cost $250 and his

hotel will be $130 each night. If he travels to Philadelphia, his flight will cost $130 and his

hotel will be $190 each night. How many nights would he need to stay in each city for the

cost to be equal for both trips?

Trip to San Diego: 250 � 130x

Trip to Philadelphia: 130 � 190x

130 � 190x � 250 � 130x

130 � 60x � 250

60x � 120

x � 2

If he stayed in each city for 2 nights, the trips would cost the same amount.

22. Your aunt is deciding between two different babysitters. Ashley charges $18 plus an

additional $8 per hour. Mark charges $10 plus an additional $10 per hour. For how many

hours of babysitting would both sitters cost the same?

Ashley: 18 � 8x

Mark: 10 � 10x

10 � 10x � 18 � 8x

10 � 2x � 18

2x � 8

x � 4

The two sitters cost the same amount for 4 hours of babysitting.

23. Devin is deciding between two trash pickup services. Rubbish Removers charges a monthly

fee of $48 plus an additional $0.10 per pound of garbage. Drake’s Disposal doesn’t charge

a monthly fee, but charges $0.20 per pound of garbage. For how many pounds of garbage

would the monthly cost of both companies be the same?

Rubbish Removers: 48 � 0.10x

Drake’s Disposal: 0.20x

0.20x � 48 � 0.10x

0.10x � 48

x � 480

In one month, the two companies cost the same for 480 pounds of trash pickup.

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Name _____________________________________________ Date _________________________

24. Daniela is buying a new mattress. Sleep Tight charges $800 for the mattress she likes, and

they charge a delivery fee of $2 per mile. Best Bedding has the same mattress for less at

$650, but they charge a higher delivery fee of $5 per mile. For what delivery distance is the

cost of the mattress the same for both furniture stores?

Sleep Tight: 800 � 2x

Best Bedding: 650 � 5x

650 � 5x � 800 � 2x

650 � 3x � 800

3x � 150

x � 50

The total costs are the same if the delivery distance is 50 miles.

25. Ken can choose between two different cell phone plans. Nationtalk Wireless charges $45

per month and gives 500 free minutes. They charge $0.02 for each additional minute. Virtual

Talk’s plan charges $50 per month and gives 600 free minutes, but charges $0.30 for each

additional minute. How many minutes would Ken need to use both plans for his monthly bill

to be the same?

Nationtalk Wireless: 45 � 0.20(x � 500)

Virtual Talk: 50 � 0.30(x � 600)

50 � 0.30(x � 600) � 45 � 0.20(x � 500)

50 � 0.30x � 180 � 45 � 0.20x � 100

0.30x � 130 � 0.20x � 55

0.10x � 130 � �55

0.10x � 75

x � 750

If Ken used both plans for 750 minutes, his monthly bill would be the same for both companies.

26. Jenna and Michelle are in a race. Jenna gives Michelle a 10-meter head start. They both

start the race at the same time. If Jenna runs 5 meters in one second, and Michelle runs

4 meters in one second, how many seconds after they start running will they be at the same

position?

Jenna: 5x

Michelle: 4x � 10

5x � 4x � 10

x � 10

After 10 seconds, they will be at the same position.


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Name _____________________________________________ Date _________________________

Rules of SportsSolving Absolute Value Equations and Inequalities

Vocabulary

Complete each sentence with a term from the box.

absolute value opposite tolerance

1. The opposite of a number is the number that is the same distance from zero but

on the other side of zero on a number line.

2. The absolute value of a number is the distance between zero and the point that

represents the number on a number line.

3. A tolerance is the amount by which a quantity is allowed to vary from the normal

or target quantity.

Problem Set Determine the absolute value of each number.

1. |0| 2. |1|

0 1

3. |�50| 4. |�102|

50 102

5. |48| 6. |�63|

48 63

7. |�91| 8. |37|

91 37

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Calculate the distance between the numbers by writing and simplifying an absolute value expression.

9. Distance between �9 and 8

|�9 � 8| � |�17| � 17

10. Distance between 11 and 4

|11 � 4| � |7| � 7

11. Distance between �3 and �12

|�3 � (�12)| � |9| � 9


|�15 � 31| � |�46| � 46

13. Distance between 6 and 12

|6 � 12| � |�6| � 6


|�23 � (�51)| � |28| � 28


|�81 � (�15)| � |�66| � 66


|�30 � (7)| � |�37| � 37

Solve the absolute value equation.

17. |x � 19| � 5

x � 19 � 5 or x � 19 � �5

x � 24 x � 14

18. |14 � x| � 30

14 � x � 30 or 14 � x � �30

x � 16 x � �44

19. |6x � 3| � 18

6x � 3 � 18 or 6x � 3 � �18

6x � 15 6x � �21

x � 15 ___ 6 � 5 __

2 x � � 21 ___

6 � � 7 __

2


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Name _____________________________________________ Date _________________________

20. |5x � 20| � 0

5x � 20 � 0

5x � 20

x � 4

21. |7x � 2| � 9 � 28

|7x � 2| � 37

7x � 2 � 37 or 7x � 2 � �37

7x � 35 7x � �39

x � 5 x � � 39 ___ 7

22. |2x � 16| � 6 � 26

|2x � 16| � 20

2x � 16 � 20 or 2x � 16 � �20

2x � 36 2x � �4

x � 18 x � �2

23. | x __ 2 � 12 | � 3 � 30

| x __ 2 � 12 | � 33

x __ 2 � 12 � 33 or x __

2 � 12 � �33

x __ 2 � 21 x __

2 � �45

x � 42 x � �90

24. | x __ 5 � 9 | � 2 � 43

| x __ 5 � 9 | � 41

x __ 5 � 9 � 41 or x __

5 � 9 � �41

x __ 5 � 32 x __

5 � �50

x � 160 x � �250

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Solve each absolute value inequality and graph your solution on a number line.

25. |x � 8| < 6

�6 < x � 8 < 6

2 < x < 14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

26. |x � 5| < 9

�9 < x � 5 < 9

�14 < x < 4

–14–13–12–11–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

27. |4x � 21| ≥ 3

4x � 21 ≥ 3 or 4x � 21 ≤ �3

4x ≥ 24 4x ≤ 18

x ≥ 6 x ≤ 18 ___ 4 � 9 __

2 � 4 1 __

2

x ≤ 4 1 __ 2 or x ≥ 6

3 4 5 6 7 8 9 10 11 12 13

28. |8x � 14| ≥ 10

8x � 14 ≥ 10 or 8x � 14 ≤ �10

8x ≥ 24 8x ≤ 4

x ≥ 3 x ≤ 4 __ 8 � 1 __

2

x ≤ 1 __ 2 or x ≥ 3

–2 –1 0 1 2 3 4 5

29. | x __ 4 | � 9 ≤ 11

| x __ 4 | ≤ 2

�2 ≤ x __ 4 ≤ 2

�8 ≤ x ≤ 8

–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11


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Name _____________________________________________ Date _________________________

30. | x __ 3 | � 24 ≤ �23

| x __ 3 | ≤ 1

�1 ≤ x __ 3 ≤ 1

�3 ≤ x ≤ 3

–4 –3 –2 –1 0 1 2 3 4

31. | x __ 6 � 1 | > 2

x __ 6 � 1 > 2 or x __

6 � 1 < �2

x __ 6 > 3 x __

6 < �1

x > 18 x < �6

x < �6 or x > 19

–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

32. | x __ 7 � 3 | > 1

x __ 7 � 3 > 1 or x __

7 � 3 < �1

x __ 7 > �2 x __

7 < �4

x > �14 x < �28

x < �28 or x > �14

–30–29–28–27–26–25–24–23–22–21–20–19–18–17–16–15–14–13–12–11

Documents

Skills Practice - Mrs. Cruz's Algebra Classmrscruzalgebra.weebly.com/uploads/8/9/5/6/8956636/c4_review...368 Chapter 4 l Skills Practice 26. An employee at a clothing store can fold