LESSON Review for Mastery x-x1-x1-4 Solving Two...

Name ________________________________________ Date __________________ Class__________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Algebra 1

Review for Mastery Solving Two-Step and Multi-Step Equations

When solving multi-step equations, first combine like terms on each side if possible. Then use inverse operations.

Solve 2x 7 3x 13. Check: 2x 3x 7 13 Group like terms together. 2x 7 3x 13

5x 7 13 Add like terms. 2(4) 7 3(4) ? 13 5x 7 13 x is multiplied by 5. Then 7 is subtracted.

7 7 Add 7 to both sides. 8 7 12 ? 13

5x 20 13 ? 13 9

5x5

205

Divide both sides by 5.

x 4

Solve each equation. Check your answers.

1. 3x 8 4 2. b2

4 26

_________________________________________ ________________________________________

3. 5y 4 2y 9 4. 14 3(x 2) 5

_________________________________________ ________________________________________

Operations Solve using Inverse Operations

4x 3 15 �• x is multiplied by 4. �• Then 3 is subtracted.

�• Add 3 to both sides. �• Then divide both sides by 4.

x3

2 9 �• x is divided by 3. �• Then 2 is added.

�• Add 2 to both sides. �• Then multiply both sides by 3.

The order of the inverse operations is the order of operations in reverse.

Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving Two-Step and Multi-Step Equations continued

A two-step equation with fractions can be simplified by multiplying each side by the LCD. This will clear the fractions.

Solve x4

23

2. Check:

x4

23

2 x4

23

2

12

x4

23

§

©¨

·

¹¸ (12)2 Multiply both sides by the LCD 12.

14

x 23

2

12

x4§

©¨

·

¹¸ 12

23§

©¨

·

¹¸ 12(2)

14

163

§

©¨

·

¹¸

23

? 2

3x 8 24 x is multiplied by 3. 8 is added. 1612

23

? 2

8 8 Add 8 to both sides. 43

23

? 2

3x 16 63

? 2

3x3

163

Divide both sides by 3. 2 ? 2 9

x 163


5. x2

38

1 6. w3

25

115

7. 3 a5

12

________________________ _________________________ ________________________

1-30

LESSON

x-xLESSON

1-x

1-30

LESSON

1-4

CS10_A1_MECR710532_C01L04d.indd 30 3/30/11 3:12:12 AM

Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving Two-Step and Multi-Step Equations

When solving multi-step equations, first combine like terms on each side if possible. Then use inverse operations.

Solve 2x 7 3x 13. Check: 2x 3x 7 13 Group like terms together. 2x 7 3x 13

5x 7 13 Add like terms. 2(4) 7 3(4) ? 13 5x 7 13 x is multiplied by 5. Then 7 is subtracted.

7 7 Add 7 to both sides. 8 7 12 ? 13

5x 20 13 ? 13 9

5x5

205

Divide both sides by 5.

x 4


1. 3x 8 4 2. b2

4 26

_________________________________________ ________________________________________

3. 5y 4 2y 9 4. 14 3(x 2) 5

_________________________________________ ________________________________________


4x 3 15 �• x is multiplied by 4. �• Then 3 is subtracted.

�• Add 3 to both sides. �• Then divide both sides by 4.

x3

2 9 �• x is divided by 3. �• Then 2 is added.

�• Add 2 to both sides. �• Then multiply both sides by 3.


Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving Two-Step and Multi-Step Equations continued

A two-step equation with fractions can be simplified by multiplying each side by the LCD. This will clear the fractions.

Solve x4

23

2. Check:

x4

23

2 x4

23

2

12

x4

23

§

©¨

·

¹¸ (12)2 Multiply both sides by the LCD 12.

14

x 23

2

12

x4§

©¨

·

¹¸ 12

23§

©¨

·

¹¸ 12(2)

14

163

§

©¨

·

¹¸

23

? 2

3x 8 24 x is multiplied by 3. 8 is added. 1612

23

? 2

8 8 Add 8 to both sides. 43

23

? 2

3x 16 63

? 2

3x3

163


x 163


5. x2

38

1 6. w3

25

115

7. 3 a5

12

________________________ _________________________ ________________________

1-31

LESSON

x-xLESSON

1-x

1-31

LESSON

1-4




Practice C 1. r 3 2. w 14

3. y 4 4. f 5

5. p 10 6. r 7

7. y 27 8. h 7

8

9. m 3 10. v 1

2

11. b 7 12. n 5

8

13. 10 14. 12

15. 30

16. 0.75x 18.50 24.25, 57 cookies

Review for Mastery 1. 4 2. 60

3. 5

3 4. 5

5. 5

4 6. 1

7. 25

2

Challenge 1. 4 inches 2. 8 inches 3. 1 inch 4. 9 inches 5. 15 inches 6. 7 inches 7. 3 inches 8. 7 inches

9. 6 inches 10. 11 inches


13. 5 sides 14. 12 sides


Problem Solving 1. 39.95 0.99d 55.79; 16 DVDs

2. 12.6 4p 1; 2.9 million 3. 9 years old 4. 10 weeks 5. D 6. F 7. B

Reading Strategies 1. like terms 2. subtraction 3. Subtract 3 from both sides, then multiply

by 5.

4. n 5 5. d 7

6. j 2

SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES

Practice A 1. 2a; 3; 10; 5 2. 4r; 9; 4

3. 5b; 30; 5b; 3b; 10 4. c 19

5. all real numbers 6. no solution

7. a. 3 hours 8. a. 2 hours

b. 75° F b. $12

Practice B 1. d 25 2. n 1

3. p 4 4. t 12

5. x 1

2 6. r 2

7. y 18 8. all real numbers

9. m 3 10. no solution

11. b 3

2 12. r 3

13. a. 8 years 14. a. 6 months

b. $52,000 b. 128 stamps

Practice C 1. x 2 2. a 3

3. c 1 4. y 3

5. d 11

2 6. t 2

7. m 9 8. all real numbers

9. no solution 10. all real numbers

11. 6 shirts, $61 12. 5 weeks, $85

Review for Mastery 1. Possible answers: add 3x to each side,

add 7x to each side

2. Possible answers: add 4x to each side,

add 10x to each side



9. m 23

10. 9x 72; x 8

11. 3x

2.8; x 8.4 12. 23

x 14; x 21

13. 8x 55.2; 6.9 mm

14. 0.23x 2.76; 12 oz

15. 6x

34,250; 205,500 points

Review for Mastery 1. divided ; multiply ; 14

2. multiplied ; divide ; 8

3. 10 4. 35

5. 7 6. 52

7. 75

8. 17

9. 12 10. 10

11. 9

10

Challenge 1. x 3 2. x 5

3. x 3 4. t 65

5. w 12

6. d 30

7. ax b; 1a

�• ax 1a

�• b; 1 �• x 1a

�• b;

x ba

8. axa

ba

; 1x ba

; x ba

9. Using the Multiplication Property of Equality, multiply each side of the equation by the reciprocal of the coefficient of the x-term. Using the Division Property of Equality, divide each side of the equation by the coefficient of the x-term. By either method, the solution is the same.

Problem Solving 1. 32c 480; $15

2. 4x 10; 2.5 grams 3. 10.50h 147; 14 hours

4. 15

w 61.50; $307.50

5. A 6. J 7. B

Reading Strategies 1. less 2. greater 3. greater 4. b 4 5. c 18 6. k 20

SOLVING TWO-STEP AND MULTI-STEP EQUATIONS

Practice A 1. 2; 10; 2 2. 3; 8; 4

3. 21; 9; 3 4. t 2

5. x 5.4 6. r 23

7. y 3 8. b 24

9. m 18

10. x 6

11. y 3 12. d 1

13. 8 14. 7x 6 5x 90

15. x 7

Practice B 1. x 1 2. y 4

3. p 7 4. m 1

5. g 8 6. h 6

7. y 50 8. n 13

9. t 13

10. x 3

11. b 2 12. q 3

13. 4 14. 5

15. 3x 5 2x 90; 19

16. 20 minutes

A5

x-5

A5

1-5

CS10_A1_MECR710532_CH01_AK.indd 5 3/29/11 12:37:06 PM

Name ________________________________________ Date __________________ Class__________________



Review for Mastery Variables and Expressions

To translate words into algebraic expressions, find words like these that tell you the operation. �• add subtract multiply divide sum difference product quotient more less times split increased decreased per ratio

Kenny owns v video games. Stan owns 7 more video games than Kenny. Write an expression for the number of video games Stan owns. v represents the number of video games Kenny owns. v 7 Think: The word �“more�” indicates addition. Order does not matter for addition. The expression 7 v is also correct.

Jenny is 12 years younger than Candy. Write an expression for Jenny’s age if Candy is c years old. c represents Candy’s age. The word “younger” means “less,” which indicates subtraction. c 12 Think: Candy is older, so subtract 12 from her age. Order does matter for subtraction. The expression 12 c is incorrect.

1. Jared can type 35 words per minute. Write an expression for the number of words he can type in m minutes. ____________________________________

2. Mr. O’Brien’s commute to work is 0.5 hour less than Miss Santos’s commute. Write an expression for the length of Mr. O’Brien’s commute if Miss Santos’s commute is h hours. ____________________________________

3. Mrs. Knighten bought a box of c cookies and split them evenly between the 25 students in her classroom. Write an expression for the number of cookies each student received. ____________________________________

4. Enrique collected 152 recyclable bottles, and Latasha collected b recyclable bottles. Write an expression for the number of bottles they collected altogether. ____________________________________

5. Tammy’s current rent is r dollars. Next month it will be reduced by $50. Write an expression for next month’s rent in dollars. ____________________________________

Name ________________________________________ Date __________________ Class__________________



Review for Mastery Variables and Expressions continued

The value of 9 depends on what number is placed in the box.

Evaluate 9 when 20 is placed in the box.

9

20 9

11

In algebra, variables are used instead of boxes.

Evaluate x 7 for x 28. x 7

28 7

4

Sometimes, the expression has more than one variable.

Evaluate x y for x 6 and y 2. x y

6 2

8

Evaluate 5 when each number is placed in the box. 6. 3 7. 5 8. 24

________________________ _________________________ ________________________

Evaluate each expression for x 4, y 6, and z 3. 9. x 15 10. 3y 11. 15 z

________________________ _________________________ ________________________

Evaluate each expression for x 2, y 18, and z 9. 12. x �• z 13. y x 14. y z

________________________ _________________________ ________________________

15.

yx 16. xy 17. z x

________________________ _________________________ ________________________

1-6

LESSON

x-xLESSON

1-x

1-6

LESSON

1-1


Name ________________________________________ Date __________________ Class__________________



Review for Mastery Variables and Expressions

To translate words into algebraic expressions, find words like these that tell you the operation. �• add subtract multiply divide sum difference product quotient more less times split increased decreased per ratio

Kenny owns v video games. Stan owns 7 more video games than Kenny. Write an expression for the number of video games Stan owns. v represents the number of video games Kenny owns. v 7 Think: The word �“more�” indicates addition. Order does not matter for addition. The expression 7 v is also correct.

Jenny is 12 years younger than Candy. Write an expression for Jenny’s age if Candy is c years old. c represents Candy’s age. The word “younger” means “less,” which indicates subtraction. c 12 Think: Candy is older, so subtract 12 from her age. Order does matter for subtraction. The expression 12 c is incorrect.

1. Jared can type 35 words per minute. Write an expression for the number of words he can type in m minutes. ____________________________________

2. Mr. O’Brien’s commute to work is 0.5 hour less than Miss Santos’s commute. Write an expression for the length of Mr. O’Brien’s commute if Miss Santos’s commute is h hours. ____________________________________

3. Mrs. Knighten bought a box of c cookies and split them evenly between the 25 students in her classroom. Write an expression for the number of cookies each student received. ____________________________________

4. Enrique collected 152 recyclable bottles, and Latasha collected b recyclable bottles. Write an expression for the number of bottles they collected altogether. ____________________________________

5. Tammy’s current rent is r dollars. Next month it will be reduced by $50. Write an expression for next month’s rent in dollars. ____________________________________

Name ________________________________________ Date __________________ Class__________________



Review for Mastery Variables and Expressions continued

The value of 9 depends on what number is placed in the box.

Evaluate 9 when 20 is placed in the box.

9

20 9

11

In algebra, variables are used instead of boxes.

Evaluate x 7 for x 28. x 7

28 7

4

Sometimes, the expression has more than one variable.

Evaluate x y for x 6 and y 2. x y

6 2

8

Evaluate 5 when each number is placed in the box. 6. 3 7. 5 8. 24

________________________ _________________________ ________________________

Evaluate each expression for x 4, y 6, and z 3. 9. x 15 10. 3y 11. 15 z

________________________ _________________________ ________________________

Evaluate each expression for x 2, y 18, and z 9. 12. x �• z 13. y x 14. y z

________________________ _________________________ ________________________

15.

yx 16. xy 17. z x

________________________ _________________________ ________________________

1-7

LESSON

x-xLESSON

1-x

1-7

LESSON

1-1




Answer Key For Equations

VARIABLES AND EXPRESSIONS

Practice A 1. the sum of a and 3 2. 2 times x 3. y less than 5 4. the quotient of n and 4 5. 10 increased by t 6. the product of 3 and s 7. c 2 8. 5m 9. 8 10. 4 11. 12 12. 3 13. 16 14. 8 15. a. j 4 b.11 years old; 16 years old; 54 years old

Practice B 1. the difference of 15 and b; b less than 15 2. the quotient of x and 16; x divided by 16 3. the sum of x and 9; 9 more than x 4. the product of 2 and t; 2 times t 5. the difference of z and 7; 7 less than z 6. the product of 4 and y; 4 times y 7. g 6 8. 10m 9. 10 10. 4 11. 7 12. 3 13. 16 14. 3 15. a. d 20 b. 40 dollars; 80 dollars; 95 dollars

Practice C 1. the sum of k and 2.5; 2.5 more than k 2. the product of 5 and n; 5 times n 3. the quotient of b and 25; b divided by 25 4. the difference of 100 and x; x less than 100 5. 35g 6. 150 p 7. 14 8. 56

9. 17

10. a. 25 2g b. 23 tokens; 17 tokens; 5 tokens; 1

token 11. n 8, Possible answer:

Tammy is 8 years younger than Nate, who is n years old

12. 4x, Possible answer: Bo had 4 times as many votes as Scott, who had x votes

Review for Mastery 1. 35m 2. h 0.5 3. c 25 4. 152 b 5. r 50 6. 8 7. 10 8. 29 9. 19 10. 18 11. 12 12. 18 13. 16 14. 2 15. 9 16. 36 17. 7

Challenge 1. 70 0.35(m)

Expression for Plan A Cost of Plan A

40 0.45(200) $130 40 0.45(300) $175 40 0.45(400) $220 40 0.45(500) $265

Expression for Plan B Cost of

Plan B 70 0.35(100) $105 70 0.35(200) $140 70 0.35(300) $175 70 0.35(400) $210 70 0.35(500) $245

2. Plan A 3. Plan B

A1

x-1

A1

1-1


Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving Equations with Variables on Both Sides

Variables must be collected on the same side of the equation before the equation can be solved. Solve 10x 2x 16. Check: 10x 2x 16 10x 2x 16

2x 2x Add 2x to both sides. 10( 2) ? 2( 2) 16

8x 16 20 ? 4 16

8x8

168


x 2 Solve 3x 5(x 2). Check: 3x 5x 10 Distribute. 3x 5(x 2)

5x 5x Add –5x to both sides. 3( 5) ? 5( 5 2)

2x 10 15 ? 5( 3)

2x2

10

2 Divide both sides by 2. 15 ? 15 9

x 5

Write the first step you would take to solve each equation.

1. 3x 2 7x 2. 4x 6 10x 3. 15x 7 3x

Solve each equation. Check your answers. 4. 4x 2 5(x 10) 5. 10 y 3 4y 13 6. 3(t 7) 2 6t 2 2t

________________________ _________________________ ________________________

Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving Equations with Variables on Both Sides continued

Some equations have infinitely many solutions. These equations are true for all values of the variable. Some equations have no solutions. There is no value of the variable that will make the equation true. Solve 3x 9 4x 9 7x. Check any value of x: 3x 9 3x 9 Combine like terms. Try x 4. 3x 3x Add 3x to each side. 3x 9 4x 9 7x

9 9 9 True statement. 3(4) 9 ? 4(4) 9 7(4)

The solution is the set of all real numbers. 12 9 ? 16 9 28

3 ? 3 9

Solve 2x 6 3x 5x 10. Check any value of x: 2x 6 3x 5x 10 Try x 1. 5x 6 5x 10 Combine like terms. 2x 6 3x 5x 10

5x 5x Add 5x to each side. 2(1) 6 3(1) ? 5(1) 10

6 10 8 False statement. 2 6 3 ? 5 10

There is no solution. 11 ? 5 8

Solve each equation.

7. x 2 x 4 8. 2x 8 2x 4 9. 5 3g 3g 5

________________________ ________________________ ________________________

10. 5x 1 4x x 7 11. 2(f 3) 4f 6 6f 12. 3x 7 2x 4x 10

________________________ ________________________ ________________________

1-38

LESSON

x-xLESSON

1-x

1-38

LESSON

1-5

CS10_A1_MECR710532_C01L05d.indd 38 3/29/11 3:46:06 PM

Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving Equations with Variables on Both Sides

Variables must be collected on the same side of the equation before the equation can be solved. Solve 10x 2x 16. Check: 10x 2x 16 10x 2x 16

2x 2x Add 2x to both sides. 10( 2) ? 2( 2) 16

8x 16 20 ? 4 16

8x8

168


x 2 Solve 3x 5(x 2). Check: 3x 5x 10 Distribute. 3x 5(x 2)

5x 5x Add –5x to both sides. 3( 5) ? 5( 5 2)

2x 10 15 ? 5( 3)

2x2

10

2 Divide both sides by 2. 15 ? 15 9

x 5

Write the first step you would take to solve each equation.

1. 3x 2 7x 2. 4x 6 10x 3. 15x 7 3x

Solve each equation. Check your answers. 4. 4x 2 5(x 10) 5. 10 y 3 4y 13 6. 3(t 7) 2 6t 2 2t

________________________ _________________________ ________________________

Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving Equations with Variables on Both Sides continued

Some equations have infinitely many solutions. These equations are true for all values of the variable. Some equations have no solutions. There is no value of the variable that will make the equation true. Solve 3x 9 4x 9 7x. Check any value of x: 3x 9 3x 9 Combine like terms. Try x 4. 3x 3x Add 3x to each side. 3x 9 4x 9 7x

9 9 9 True statement. 3(4) 9 ? 4(4) 9 7(4)

The solution is the set of all real numbers. 12 9 ? 16 9 28

3 ? 3 9

Solve 2x 6 3x 5x 10. Check any value of x: 2x 6 3x 5x 10 Try x 1. 5x 6 5x 10 Combine like terms. 2x 6 3x 5x 10

5x 5x Add 5x to each side. 2(1) 6 3(1) ? 5(1) 10

6 10 8 False statement. 2 6 3 ? 5 10

There is no solution. 11 ? 5 8


7. x 2 x 4 8. 2x 8 2x 4 9. 5 3g 3g 5

________________________ ________________________ ________________________

10. 5x 1 4x x 7 11. 2(f 3) 4f 6 6f 12. 3x 7 2x 4x 10

________________________ ________________________ ________________________

1-39

LESSON

x-xLESSON

1-x

1-39

LESSON

1-5




Practice C 1. r 3 2. w 14

3. y 4 4. f 5

5. p 10 6. r 7

7. y 27 8. h 7

8

9. m 3 10. v 1

2

11. b 7 12. n 5

8

13. 10 14. 12

15. 30

16. 0.75x 18.50 24.25, 57 cookies

Review for Mastery 1. 4 2. 60

3. 5

3 4. 5

5. 5

4 6. 1

7. 25

2

Challenge 1. 4 inches 2. 8 inches 3. 1 inch 4. 9 inches 5. 15 inches 6. 7 inches 7. 3 inches 8. 7 inches





Problem Solving 1. 39.95 0.99d 55.79; 16 DVDs

2. 12.6 4p 1; 2.9 million 3. 9 years old 4. 10 weeks 5. D 6. F 7. B

Reading Strategies 1. like terms 2. subtraction 3. Subtract 3 from both sides, then multiply

by 5.

4. n 5 5. d 7

6. j 2

SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES

Practice A 1. 2a; 3; 10; 5 2. 4r; 9; 4

3. 5b; 30; 5b; 3b; 10 4. c 19

5. all real numbers 6. no solution

7. a. 3 hours 8. a. 2 hours

b. 75° F b. $12

Practice B 1. d 25 2. n 1

3. p 4 4. t 12

5. x 1

2 6. r 2

7. y 18 8. all real numbers

9. m 3 10. no solution

11. b 3

2 12. r 3

13. a. 8 years 14. a. 6 months

b. $52,000 b. 128 stamps

Practice C 1. x 2 2. a 3

3. c 1 4. y 3

5. d 11

2 6. t 2

7. m 9 8. all real numbers

9. no solution 10. all real numbers

11. 6 shirts, $61 12. 5 weeks, $85

Review for Mastery 1. Possible answers: add 3x to each side,

add 7x to each side

2. Possible answers: add 4x to each side,

add 10x to each side



9. m 23

10. 9x 72; x 8

11. 3x

2.8; x 8.4 12. 23

x 14; x 21

13. 8x 55.2; 6.9 mm

14. 0.23x 2.76; 12 oz

15. 6x

34,250; 205,500 points

Review for Mastery 1. divided ; multiply ; 14

2. multiplied ; divide ; 8

3. 10 4. 35

5. 7 6. 52

7. 75

8. 17

9. 12 10. 10

11. 9

10

Challenge 1. x 3 2. x 5

3. x 3 4. t 65

5. w 12

6. d 30

7. ax b; 1a

�• ax 1a

�• b; 1 �• x 1a

�• b;

x ba

8. axa

ba

; 1x ba

; x ba

9. Using the Multiplication Property of Equality, multiply each side of the equation by the reciprocal of the coefficient of the x-term. Using the Division Property of Equality, divide each side of the equation by the coefficient of the x-term. By either method, the solution is the same.

Problem Solving 1. 32c 480; $15

2. 4x 10; 2.5 grams 3. 10.50h 147; 14 hours

4. 15

w 61.50; $307.50

5. A 6. J 7. B

Reading Strategies 1. less 2. greater 3. greater 4. b 4 5. c 18 6. k 20

SOLVING TWO-STEP AND MULTI-STEP EQUATIONS

Practice A 1. 2; 10; 2 2. 3; 8; 4

3. 21; 9; 3 4. t 2

5. x 5.4 6. r 23

7. y 3 8. b 24

9. m 18

10. x 6

11. y 3 12. d 1

13. 8 14. 7x 6 5x 90

15. x 7

Practice B 1. x 1 2. y 4

3. p 7 4. m 1

5. g 8 6. h 6

7. y 50 8. n 13

9. t 13

10. x 3

11. b 2 12. q 3

13. 4 14. 5

15. 3x 5 2x 90; 19

16. 20 minutes

A5

x-5

A5

1-5




Review for Mastery

1. s 4P 2. b 180 a – c

3. K VPT

4. w 3Vlh

5. 12 in. 6. add –x to both sides 7. multiply both sides by 2 8. add 3r to both sides

9. a 3cb

10. z 3(y x)

11. m pn 3

Challenge 1. l 375 2. P 2000 3. t 5 years 4. r 0.032 or 3.2% 5. $351 6. 3 years 7. $2200 8. 0.03 or 3% 9. $247 10. 5 years 11. 0.027 or 2.7%

Problem Solving

1. r dt

2. 9.1 m/s

3. 8.5 m/s 4. 0.4 m/s 5. B 6. F 7. D 8. J

Reading Strategies 1. Possible answer: 3x 2y 9 2. The equation contains only one

variable, n. 3. Yes, because it has two or more

variables. 4. Divide both sides by r.

5. t 83

b

6. a. h VIw

b. 3 cm

SOLVING ABSOLUTE-VALUE EQUATIONS

Practice A 1. 3; 2; 2; 2 2. 7; 7; 4; 4; 4; 4; 11; 3 3. 6; 6; 6; 5; 7 4. { 8, 8} 5. { 14, 14} 6. { 9, 9} 7. { 17, 17} 8. { 11, 7} 9. { 1, 11} 10. { 5, 5} 11. { 7, 3} 12. { 11, 9}

13. x 24 2

14. 22 miles per gallon; 26 miles per gallon

Practice B 1. { 12, 12}

2. 12

,12

®¯

½¾¿

3. { 10, 10} 4. { 9, 9} 5. { 8, 8} 6. { 13, 7} 7. { 1, 3} 8. {2, 8} 9. { 14, 10} 10. { 3, 3} 11. {1} 12. {3} 13. two 14. one

15. none 16. x 68 3.5

14. 64.5 ; 71.5

Practice C

1. 35

,35

®¯

½¾¿

2. {0}

3. { 10.5, 10.5} 4. 5. { 7, 7} 6. { 11}

7. 32

,52

®¯

½¾¿

8.



3. Possible answers: add 3x to each side, add 15x to each side

4. 48 5. 2 6. 5 7. no solution 8. 1 9. all real numbers 10. no solution 11. all real numbers 12. 1

Challenge

1. x 73

2. x 4

3. x 7 12

4. x 2

5. 2x ( 6) 0 or 2x 6 0 6. 9x 18 0 or 9x ( 18) 0 7. 5x 27 0 or 5x ( 27) 0 8. 3x ( 21) 0 or 3x 21 0

9. a. x ba

b. After writing the equation in the form ax b 0, substitute the values of a and b into the formula found in part a.

10. x 3 11. x 2

12. x 5 25

13. x 7

14. a. infinitely many solutions b. no solution

Problem Solving 1. 28 feet 2. 7 days 3. 16 hours 4. 20 months 5. B 6. G 7. C

Reading Strategies 1. Look for an opportunity to use the

Distributive Property. 2. Combine any like terms. 3. Collect the variable terms on one side of

the equal sign. 4. p 6 5. x 1 6. t 2

SOLVING FOR A VARIABLE

Practice A

1. C K 273 2. f 1T

3. y 5x 4. s r 4t

5. m 73

p n 6. j 6 kh

7. w 9v 8. a bc 3

9. a. t dr

10. a. F 2 E V

b. 3 b. 12

Practice B

1. r 2C 2. m y b

x

3. c 4d 4. n 8 6m

5. p 52

q r 6. x 10 zy

7. b ac

8. j 4hk

9. a. p 2155

c 10. a. b 2Ah

b. 17 b. 32 mm

Practice C

1. w 22

P l 2. vf at vi

3. f 3

g 4. a 12 5b

5. x 73

z y 6. h 5

jk g

7. r s(t 9) 8. n 3mp

9. a. a Fm

10. a. t IPr

b. 19 m/s2 b. 8 years

A6

x-6

A6

1-6


Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving for a Variable

Solving for a variable in a formula can make it easier to use that formula. The process is similar to that of solving multi-step equations. Find the operations being performed on the variable you are solving for, and then use inverse operations.

The formula A 12

bh relates the area A of a triangle

to its base b and height h. Solve the formula for b.

A 12

bh b is multiplied by 12

.

21§

©¨

·

¹¸ �• A

21§

©¨

·

¹¸ 12

bh Multiply both sides by 21

.

2A bh b is multiplied by h.

2Ah

bhh

Divide both sides by h.

2Ah

b Simplify.

Solve for the indicated variable.

1. P 4s for s 2. a b c 180 for b 3. P KTV

for K

________________________ _________________________ ________________________

The formula V 13

lwh relates the volume of a square pyramid

to its base length l, base width w, and height h.

4. Solve the formula for w. ___________________________

5. A square pyramid has a volume of 560 in3, a base length of 10 in., and a height of 14 in. What is its base width? ___________________________


A lw Solve for w.

�• w is multiplied by l. �• Divide both sides by l.

P 2l 2w Solve for w.

�• w is multiplied by 2. �• Then 2l is added.

�• Add 2l to both sides. �• Then divide both sides by 2.


Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving for a Variable continued

Any equation with two or more variables can be solved for any given variable.

Solve x y z10 for y.

x y z10

y z is divided by 10.

10(x) 10

y z10

§

©¨

·

¹¸ Multiply both sides by 10.

10x y z z is subtracted from y. Add z to both sides. z z

10x z y

Solve a b cd

for c.

a b cd

b b Add b to each side.

a b cd

d(a b)

cd§

©¨

·

¹¸d Multiply both sides by d.

d(a b) c Simplify.

State the first inverse operation to perform when solving for the indicated variable.

6. y x z; for z _________________________________________________________

7. f g

2 h; for g _________________________________________________________

8. t 3r s5

; for s _________________________________________________________


9. 3ab c; for a 10. y x z3

; for z 11. m 3n

p; for m

________________________ _________________________ ________________________

LESSON

x-x

1-461-46

LESSON

1-6


Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving for a Variable

Solving for a variable in a formula can make it easier to use that formula. The process is similar to that of solving multi-step equations. Find the operations being performed on the variable you are solving for, and then use inverse operations.

The formula A 12

bh relates the area A of a triangle

to its base b and height h. Solve the formula for b.

A 12

bh b is multiplied by 12

.

21§

©¨

·

¹¸ �• A

21§

©¨

·

¹¸ 12

bh Multiply both sides by 21

.

2A bh b is multiplied by h.

2Ah

bhh

Divide both sides by h.

2Ah

b Simplify.


1. P 4s for s 2. a b c 180 for b 3. P KTV

for K

________________________ _________________________ ________________________

The formula V 13

lwh relates the volume of a square pyramid

to its base length l, base width w, and height h.

4. Solve the formula for w. ___________________________

5. A square pyramid has a volume of 560 in3, a base length of 10 in., and a height of 14 in. What is its base width? ___________________________


A lw Solve for w.

�• w is multiplied by l. �• Divide both sides by l.

P 2l 2w Solve for w.

�• w is multiplied by 2. �• Then 2l is added.

�• Add 2l to both sides. �• Then divide both sides by 2.


Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving for a Variable continued

Any equation with two or more variables can be solved for any given variable.

Solve x y z10 for y.

x y z10

y z is divided by 10.

10(x) 10

y z10

§

©¨

·

¹¸ Multiply both sides by 10.

10x y z z is subtracted from y. Add z to both sides. z z

10x z y

Solve a b cd

for c.

a b cd

b b Add b to each side.

a b cd

d(a b)

cd§

©¨

·

¹¸d Multiply both sides by d.

d(a b) c Simplify.

State the first inverse operation to perform when solving for the indicated variable.

6. y x z; for z _________________________________________________________

7. f g

2 h; for g _________________________________________________________

8. t 3r s5

; for s _________________________________________________________


9. 3ab c; for a 10. y x z3

; for z 11. m 3n

p; for m

________________________ _________________________ ________________________

LESSON

x-x

1-471-47

LESSON

1-6




Review for Mastery

1. s 4P 2. b 180 a – c

3. K VPT

4. w 3Vlh

5. 12 in. 6. add –x to both sides 7. multiply both sides by 2 8. add 3r to both sides

9. a 3cb

10. z 3(y x)

11. m pn 3

Challenge 1. l 375 2. P 2000 3. t 5 years 4. r 0.032 or 3.2% 5. $351 6. 3 years 7. $2200 8. 0.03 or 3% 9. $247 10. 5 years 11. 0.027 or 2.7%

Problem Solving

1. r dt

2. 9.1 m/s

3. 8.5 m/s 4. 0.4 m/s 5. B 6. F 7. D 8. J

Reading Strategies 1. Possible answer: 3x 2y 9 2. The equation contains only one

variable, n. 3. Yes, because it has two or more

variables. 4. Divide both sides by r.

5. t 83

b

6. a. h VIw

b. 3 cm

SOLVING ABSOLUTE-VALUE EQUATIONS

Practice A 1. 3; 2; 2; 2 2. 7; 7; 4; 4; 4; 4; 11; 3 3. 6; 6; 6; 5; 7 4. { 8, 8} 5. { 14, 14} 6. { 9, 9} 7. { 17, 17} 8. { 11, 7} 9. { 1, 11} 10. { 5, 5} 11. { 7, 3} 12. { 11, 9}

13. x 24 2

14. 22 miles per gallon; 26 miles per gallon

Practice B 1. { 12, 12}

2. 12

,12

®¯

½¾¿

3. { 10, 10} 4. { 9, 9} 5. { 8, 8} 6. { 13, 7} 7. { 1, 3} 8. {2, 8} 9. { 14, 10} 10. { 3, 3} 11. {1} 12. {3} 13. two 14. one

15. none 16. x 68 3.5

14. 64.5 ; 71.5

Practice C

1. 35

,35

®¯

½¾¿

2. {0}

3. { 10.5, 10.5} 4. 5. { 7, 7} 6. { 11}

7. 32

,52

®¯

½¾¿

8.



3. Possible answers: add 3x to each side, add 15x to each side

4. 48 5. 2 6. 5 7. no solution 8. 1 9. all real numbers 10. no solution 11. all real numbers 12. 1

Challenge

1. x 73

2. x 4

3. x 7 12

4. x 2

5. 2x ( 6) 0 or 2x 6 0 6. 9x 18 0 or 9x ( 18) 0 7. 5x 27 0 or 5x ( 27) 0 8. 3x ( 21) 0 or 3x 21 0

9. a. x ba

b. After writing the equation in the form ax b 0, substitute the values of a and b into the formula found in part a.

10. x 3 11. x 2

12. x 5 25

13. x 7

14. a. infinitely many solutions b. no solution

Problem Solving 1. 28 feet 2. 7 days 3. 16 hours 4. 20 months 5. B 6. G 7. C

Reading Strategies 1. Look for an opportunity to use the

Distributive Property. 2. Combine any like terms. 3. Collect the variable terms on one side of

the equal sign. 4. p 6 5. x 1 6. t 2

SOLVING FOR A VARIABLE

Practice A

1. C K 273 2. f 1T

3. y 5x 4. s r 4t

5. m 73

p n 6. j 6 kh

7. w 9v 8. a bc 3

9. a. t dr

10. a. F 2 E V

b. 3 b. 12

Practice B

1. r 2C 2. m y b

x

3. c 4d 4. n 8 6m

5. p 52

q r 6. x 10 zy

7. b ac

8. j 4hk

9. a. p 2155

c 10. a. b 2Ah

b. 17 b. 32 mm

Practice C

1. w 22

P l 2. vf at vi

3. f 3

g 4. a 12 5b

5. x 73

z y 6. h 5

jk g

7. r s(t 9) 8. n 3mp

9. a. a Fm

10. a. t IPr

b. 19 m/s2 b. 8 years

A7

x-7

A7

1-7


Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving Absolute-Value Equations

There are three steps in solving an absolute-value equation. First use inverse operations to isolate the absolute-value expression. Then rewrite the equation as two cases that do not involve absolute values. Finally, solve these new equations.

Solve x 3 4 = 8.

Step 1: Isolate the absolute-value expression.

x 3 4 = 8

4 4 Subtract 4 from both sides.

x 3 = 4

Step 2: Rewrite the equation as two cases.

x 3 = 4

Case 1 Case 2 Step 3: x 3 = 4 x 3 = 4 Solve. 3 3 3 3 Add 3 to both sides. x = 1 x = 7 The solution are 1 and 7.


1. x 2 3 = 5 2. x 7 2 = 10

_________________________________________ ________________________________________

3. 4 x 5 = 20 4. 2x 1 = 7

_________________________________________ ________________________________________

Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving Absolute-Value Equations continued

Some absolute-value equations have two solutions. Others have one solution or no solution. To decide how many solutions there are, first isolate the absolute-value expression.

Solve 2x 1 3 = 7.

2x 1 3 = 7

3 3 Add 3 to both sides.

2x 1 = 4 Absolute value cannot be negative.

The equation has no solution.


5. 8 x 2 = 8 6. x 1 5 = 2

_________________________________________ ________________________________________

7. 4 x 3 = 16 8. 3 x 10 = 0

_________________________________________ ________________________________________

Original Equation Simplified Equation Solutions

x 5 7 x 5 7

5 5

x 2

x 2 has two solutions, x 2 and x 2. The solutions are 2 and 2.

x 5 2 2 x 5 2 2

2 2

x 5 0

x 5 0 means x 5 0, so there is one solution x 5. The solution is 5.

x 7 4 1 x 7 4 1

4 4

x 7 3

x 7 3 has no solutions

because an absolute-value expression is never negative. There is no solution.

LESSON

x-x

1-541-54

LESSON

1-7


Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving Absolute-Value Equations

There are three steps in solving an absolute-value equation. First use inverse operations to isolate the absolute-value expression. Then rewrite the equation as two cases that do not involve absolute values. Finally, solve these new equations.

Solve x 3 4 = 8.

Step 1: Isolate the absolute-value expression.

x 3 4 = 8

4 4 Subtract 4 from both sides.

x 3 = 4

Step 2: Rewrite the equation as two cases.

x 3 = 4

Case 1 Case 2 Step 3: x 3 = 4 x 3 = 4 Solve. 3 3 3 3 Add 3 to both sides. x = 1 x = 7 The solution are 1 and 7.


1. x 2 3 = 5 2. x 7 2 = 10

_________________________________________ ________________________________________

3. 4 x 5 = 20 4. 2x 1 = 7

_________________________________________ ________________________________________

Name ________________________________________ Date __________________ Class__________________



Review for Mastery Solving Absolute-Value Equations continued

Some absolute-value equations have two solutions. Others have one solution or no solution. To decide how many solutions there are, first isolate the absolute-value expression.

Solve 2x 1 3 = 7.

2x 1 3 = 7

3 3 Add 3 to both sides.

2x 1 = 4 Absolute value cannot be negative.

The equation has no solution.


5. 8 x 2 = 8 6. x 1 5 = 2

_________________________________________ ________________________________________

7. 4 x 3 = 16 8. 3 x 10 = 0

_________________________________________ ________________________________________

Original Equation Simplified Equation Solutions

x 5 7 x 5 7

5 5

x 2

x 2 has two solutions, x 2 and x 2. The solutions are 2 and 2.

x 5 2 2 x 5 2 2

2 2

x 5 0

x 5 0 means x 5 0, so there is one solution x 5. The solution is 5.

x 7 4 1 x 7 4 1

4 4

x 7 3

x 7 3 has no solutions

because an absolute-value expression is never negative. There is no solution.

LESSON

x-x

1-551-55

LESSON

1-7




Challenge 1. bc: 7 �• 15 105; ad: 5 �• 21 105

2. a. Multiply each side by bdac

b. Cross-Product Property; Multiplication Property of Equality; Identity Property of Multiplication

3. ab

cd

ad bc

1ab

§ ·¨ ¸© ¹

ad 1ab

§ ·¨ ¸© ¹

bc

db

ca

Cross-Product Property; Multiplication Property of Equality; Identity Property of Multiplication

Problem Solving 1. 2.67 donuts/minute 2. $8800 3. 63,000 babies 4. 0.53 mi/min 5. C 6. G 7. D 8. F Reading Strategies

1. 16 oz 1 lbor1 lb 16 oz

2. no; the second quality is not 1. 3. The two quantities are equal. 4. ratio; rate; ratio 5. 28 pg/hr

6. Possible answer: 1 122 24

APPLICATIONS OF PROPORTIONS

Practice A 1. 4 2. 28

3. 5x

412

; 15 feet

4. width 4 cm; length 12 cm

5. 12

6. 16 cm; 32 cm;

12 cm2; 48 cm2

7. 12

8. They are the same. 9. 14

10. The ratio of the areas is the square of the ratio in problem 5.

Practice B 1. 15 2. 3.2

3. 5.5x

420

; 27.5 feet

4. The ratio of the volumes is the cube of the ratio of the corresponding sides.

5. 12

Practice C 1. 7.5 2. 6

3. 5.5x

1.37511

; 44 feet

4. 3 5. 14

Review for Mastery 1. 15.6 ft 2. 6.125 cm 3. 5; 25; (5)3 125

4. 23

; 22

3§ ·¨ ¸© ¹

49

; 827

5. 107

; 107

; 107

; 310

7§ ·¨ ¸© ¹

1000343

6. The ratio of the areas is the square of the ratio of corresponding dimensions:

2120

§ ·¨ ¸© ¹

1400

.

7. The ratio of the circumferences is equal to the ratio of corresponding dimensions: 4.

Challenge

1. ST ; YZ 2. 1 in.; 2 in.; 3.5 in.

3. DE ; XY 4. 2 in.; 4 in.



9. { 6.6, 8.6} 10. { 8, 3}

11. 12.

13. x 3 0.005; 2.995 m; 3.005 m

14. x 5 1.5; 6.5 C; 3.5 C

Review for Mastery 1. { 6, 10} 2. { 15, 1}

3. {0, 10} 4. { 3, 3}

5. {2} 6.

7. 8. { 10}

Challenge 1.

( 3) 2 5 5; 7 2 5 5

2. yes; possible answer: x 2 1 6

3. x 5 4

4. x 2 6

5. x 3 8

6. x 5 4

7. x 1

3

8. x 5.5 0.5

9. x 2.5 7

10. x 1

3

2

Problem Solving

1. x 70 0.02 ; 69.98 cm; 70.02 cm

2. x 53 = 0.021; 52.979 m; 53.021 m

3. 1 and 11 4. 11.9 cm 5. B 6. G 7. C

Reading Strategies 1. one

2. two 3. none

4. one

5. none

6. two

RATES, RATIOS, AND PROPORTIONS

Practice A 1. 20 2. 58 ft/s

3. $1.05/lb 4. 2.5 pages/min

5. y 4 6. x 18

7. m 2 8. t 75

9. b 4 10. x 1

11. 150 in. 12. 160 mi

Practice B 1. 15 2. $0.49/lb

3. 0.1 cars/min 4. 46.9 ft/s

5. y 5 6. x 0.4

7. m 96 8. t 5

3

9. b 20 10. x 3.5

11. 185 in. 12. 3.7 cm

Practice C 1. 18 2. 4.8 lb/book

3. $7.90/h 4. 293.3 ft/s

5. x 1.5 6. b 0.5

7. s 0.6 8. y 2

9. x 10 10. y 3

4

11. 125.5 ft 12. 3.2 in.

Review for Mastery

1. 3

1 2.

1

100

3. 8

1

4. 1000 m

1 km;

3,391,100

83.5; 40,612

5. 320 c/min 6. x 2.5

7. k 3 8. a 10.75

9. y 30

A8

x-8

A8

1-8


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