chapter foundations of algebra 1 solutions key

Foundations of AlgebraSolutions Key

Are You reAdY?

1. E 2. A

3. C 4. D

5. (7 - 3) ÷ 2 = 4 ÷ 2 = 2

6. 4 · 6 ÷ 3 = 24 ÷ 3 = 8

7. 12 - 3 + 1 = 9 + 1 = 10

8. 2 · 10 ÷ 5 = 20 ÷ 5 = 4

9. 125 ÷ 5 2 = 125 ÷ 25 = 5

10. 7 · 6 + 5 · 4 = 42 + 20 = 62

11. -15 + 19 = 4

12. -6 - (-18) = -6 + 18 = 12

13. 6 + (-8) = 6 - 8 = -2

14. -12 + (-3) = -12 - 3 = -15

15. 1 __ 4 + 2 __

3

= 3 ___ 12

+ 8 ___ 12

= 3 + 8 _____ 12

= 11 ___ 12

16. 1 1 __ 2 - 3 __

4

= 6 __ 4 - 3 __

4

= 6 - 3 _____ 4

= 3 __ 4

17. 3 __ 8 + 2 __

3

= 9 ___ 24

+ 16 ___ 24

= 9 + 16 ______ 24

= 25 ___ 24

18. 3 __ 2 - 2 __

3

= 9 __ 6 - 4 __

6

= 9 - 4 _____ 6

= 5 __ 6

19. 2x + 3 = 2(7) + 3 = 14 + 3 = 17

20. 3n - 5 = 3(7) - 5 = 21 - 5 = 16

21. 13 - 4a = 13 - 4(2) = 13 - 8 = 5

22. 3y + 5 = 3(5) + 5 = 15 + 5 = 20

23. 4(0.75) + 6(0.60) 24. 13ℓ

25. Possible answer: a number plus 2 times itself

VAriAbles And expressions

CheCk it OUt!

1. Possible answers given

a. 4 decreased by n; n less than 4

b. the sum of 9 and q; q added to 9

c. the quotient of t and 5; t divided by 5

d. the product of 3 and h; 3 times h.

2a. 65t b. m + 5

c. 32d 3a. mn = 3 · 2 = 6

b. p - n= 9 - 2 = 7

c. p ÷ m= 9 ÷ 3 = 3

4a. 63s b. 63(12) = 756 bottles;63(25) = 1575 bottles;63(50) = 3150 bottles

think and disCUss

1. addition - increase by, sum of; subtraction - decreased by, difference of; multiplication - multiplied by, product of; division - divided by, quotient of

2. Both types of expressions may contain numbers and operation. Algebraic expressions may also contain variables.

3. Words

Addition

Subtraction

Multiplication

Division

3 more than x

1 less than y

The product of 2 and n

The quotient of x and 4

x + 3

y - 1

2n

Algebra

x ÷ 4

eXeRCisesguided practice

1. variable

2–9. Possible answers given

2. 5 less than n; n decreased by 5

3. the quotient of f and 3; f divided by 3

4. c increased by 15; the sum of c and 15

5. 9 decreased by y; y less than 9

6. one-twelfth x; the quotient of x and 12

7. the sum of t and 12; t increased by 12

8. the product of 8 and x; 8 groups of x

9. x decreased by 3; the difference of x and 3

10. 45h 11. w + 4

12. a - c= 3 - 2 = 1

13. ab 3 · 4 = 12

14. b ÷ c4 ÷ 2 = 2

15. ac3 · 2 = 6

16a. 0.5d b. 0.5d 0.5(2) = 1 0.5(4) = 2 0.5(10) = 5

practice and problem Solving

17. the product of 5 and p; 5 groups of p

18. 4 decreased by y; the difference of 4 and y

19. the sum of 3 and x; 3 increased by x

20. the product of 3 and y; 3 times y

1CHAPTER

1 Holt McDougal Algebra 1

1-1

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21. negative 3 times s; the product of negative 3 and s

22. the quotient of r and 5; one-fifth r

23. 14 decreased by t; the difference of 14 and t

24. the sum of x and 0.5; x increased by 0.5

25. t + 20 26. 8n

27. 6 - 5 = 1 28. 5 + 3 = 8

29. 6 ÷ 3 = 2 30. 5 · 6 = 30

31a. h - 40

b. h - 40 (40) - 40 = 0 (44) - 40 = 0 (48) - 40 = 0 (52) - 40 = 0

32. To evaluate an expression is to find its value. To do this, substitute values for the variables and perform all the indicated operations.

33. 2x; possible answer: Jim has twice as many aunts as Carly, who has x aunts.

34. 17 - b; possible answer: Sarah started with 17 apples, but lost b of them.

35. y + 10; possible answer; April had y CDs and then got 10 more.

36a. air pressure

b. depth below the water in feet.

c. 14.7 + 0.445d = 14.7 + 0.445(8) = 14.7 + 3.56 = 18.26 psi

37. A = ℓ · wA = 9wA = 9(1) = 9 in 2 A = 9(8) = 72 in 2 A = 9(9) = 81 in 2 A = 9(11) = 99 in 2

38a. P = 2ℓ + 2w b. P = 2ℓ + 2w = 2(14) + 2(8) = 28 + 16 = 44 cm

c. A = ℓ · w or ℓw d. A = ℓw = (8)(14) = 112 cm 2

39. x x + 12

1 (1) + 12 = 13

2 (2) + 12 = 14

3 (3) + 12 = 15

4 (4) + 12 = 16

40. x 10x1 10(1) = 10

5 10(5) = 50

10 10(10) = 100

15 10(15) = 150

41. x x ÷ 2

12 (12) ÷ 2 = 6

20 (20) ÷ 2 = 10

26 (26) ÷ 2 = 13

30 (30) ÷ 2 = 15

42a. Let p represent the weight of an object on Earth in pounds; 0.38p

b. p = 120 + 44 = 164 0.38p = 0.38(164) = 62.31 lbs

43a. 47.84 + m b. 58.53 - s

44. Both algebraic and numerical expressions contain numbers and operations, but algebraic expressions also contain variables.

Verbal algebraic x = 12 x = 14x reduced

by 5 x - 5 12 - 5 = 7 14 - 5 = 9

45. 7 more than x x + 7 12 + 7 = 19 14 + 7 = 21

46. The quotient of x and 2

x _ 2 12

__ 2 = 6 14 __

2 = 7

47. The sum of x and 3 x + 3 12 + 3 = 15 14 + 3 = 17

teSt prep

48. C; b fewer than 3. 49. F; 12 - 5

50. B; Sarah has driven the difference of 25 and x.

challenge and extend

51. 2ab = 2(6)(3) 52. 2x + y = 2(4) + (5)= 36 = 8 + 5 = 13

53. 3x ÷ 6y = 3(6) ÷ 6(3)= 1

54. Let h represent the number of hours used in a month when h is more than 20 hours.9.95 + 0.50(h - 20)= 9.95 + 0.50(35 - 20)= 9.95 + 0.50(15) = $17.45

solVing equAtions bY Adding or subtrActing

CheCk it OUt!

1a. n - 3.2 = 5.6 ______ + 3.2 _____ + 3.2 n = 8.8

b. -6 = k - 6 ___ +6 _____ + 6 0 = k

c. 16 = m - 9 ___ + 9 _____ + 9 25 = m

2a. d + 1 __ 2 = 1

_____

- 1 __ 2

____ - 1 __

2

d = 1 __ 2

b. -5 = k + 5 ___ - 5 _____ - 5 -10 = k

c. 6 + t = 14 _______ - 6 ___ - 6 t = 8


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3a. -2.3 + m = 7 _________ + 2.3 _____ + 2.3 m = 9.3

b. - 3 __ 4 + z = 5 __

4

________

+ 3 __ 4

____ + 3 __

4

z = 2

c. -11 + x = 33 ________ + 11 ____ + 11 x = 44

4. a + r = 220 a + 185 = 220 _______ - 185 _____ -185 a = 35 The age of a person who has a maximum heart rate

of 185 beats per minute is 35 years old.

think and disCUss

1. Possible answer: If a scale is balanced, then you can add or remove the same weight on both sides without affecting the balance. Similarly, in an equation, the Addition and Subtraction Properties of Equality say that you can add or subtract the same value on both sides without affecting the equality.

2. Properties of Equality

x - 6 = 2 x = 8

+ 4 + x = -1

x = -5

-


1. Possible answer: The solution of an equation is a number. It is a value for the variable that works in the equation.

2. s - 5 = 3 _____ + 5 ___ + 5 s = 8

3. 17 = w - 4 ___ + 4 _____ + 4 21 = w

4. k - 8 = -7 _____ + 8 ___ + 8 k = 1

5. x - 3.9 = 12.4 ______ + 3.9 _____ + 3.9 x = 16.3

6. 8.4 = y - 4.6 _____ + 4.6 _______ + 4.6 13 = y

7. 3 __ 8 = t - 1 __

8

____

+ 1 __ 8

_____ + 1 __

8

1 __ 2 = t

8. t + 5 = -25 ____ - 5 ____ - 5 t = -30

9. 9 = s + 9 ___ - 9 _____ - 9 0 = s

10. 42 = m + 36 ____ - 36 _______ - 36 6 = m

11. 2.8 = z + 0.5 _____ - 0.5 _______ - 0.5 2.3 = z

12. b + 2 __ 3 = 2

_____

- 2 __ 3

____ - 2 __

3

b = 4 __ 3

13. n + 1.8 = 3 ______ - 1.8 _____ - 1.8 n = 1.2

14. -10 + d = 7 ________ + 10 ____ + 10 d = 17

15. 20 = -12 + v ____ + 12 ________ + 12 32 = v

16. -46 + q = 5 ________ + 46 ____ + 46 q = 51

17. 2.8 = -0.9 + y _____ + 0.9 ________ + 0.9 3.7 = y

18. - 2 __ 3 + c = 2 __

3

________

+ 2 __ 3

____ + 2 __

3

c = 4 __ 3

19. - 5 __ 6

+ p = 2

________

+ 5 __ 6

____

+ 5 __ 6

p = 17 ___ 6

20. Let w represent the weight of the original diamond. w - 45 = 67 ______ + 45 ____ + 45 w = 112 The original diamond weighed 112 carats.

Possible answer: The weight of the diamond was reduced by about 50 carats, resulting in a diamond that weighed about 70 carats. So the original weight will be close to 50 + 70 = 120 carats. So 112 carats is reasonable.


21. 1 = k - 8 ___ + 8 _____ + 8 9 = k

22. u - 15 = -8 ______ + 15 ____ + 15 u = 7

23. x - 7 = 10 _____ + 7 ___ + 7 x = 17

24. -9 = p - 2 ___ + 2 _____ + 2 -7 = p

25. 3 __ 7 = p - 1 __

7

____

+ 1 __ 7

______ + 1 __

7

4 __ 7 = p

26. q - 0.5 = 1.5 ______ + 0.5 _____ + 0.5 q = 2

27. 6 = t - 4.5 _____ + 4.5 ______ + 4.5 10.5 = t

28. 4 2 __ 3

= r - 1 __ 3

____

+ 1 __ 3

_____

+ 1 __ 3

5 = r

29. 6 = x - 3 ___ + 3 _____ + 3 9 = x

30. 1.75 = k - 0.75 ______ + 0.75 ________ + 0.75 2.50 = k

31. 19 + a = 19 ________ - 19 ____ - 19 a = 0

32. 4 = 3.1 + y _____ - 3.1 ________ - 3.1 0.9 = y

33. m + 20 = 3 ______ - 20 ____ - 20 m = -17

34. -12 = c + 3 ___ - 3 _____ - 3 -15 = c

35. v + 2300 = -800 ________ - 2300 ______ - 2300 v = -3100

36. b + 42 = 300 ______ - 42 ____ - 42 b = 258

37. 3.5 = n + 4 _____ - 4 _____ - 4 -0.5 = n

38. b + 1 __ 2

= 1 __ 2

_____

- 1 __ 2

____

- 1 __ 2

b = 0


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39. x + 5.34 = 5.39 _______ - 5.34 ______ - 5.34 x = 0.05

40. 2 = d + 1 __ 4

____

- 1 __ 4

______ - 1 __

4

7 __ 4 = d

41. -12 + f = 3 ________ + 12 ____ + 12 f = 15

42. -9 = -4 + g ___ + 4 _______ + 4 -5 = g

43. -1200 + j = 345 __________ + 1200 ______ + 1200 j = 1545

44. 90 = -22 + a ____ + 22 ________ + 22 112 = a

45. 26 = -4 + y ___ + 4 _______ + 4 30 = y

46. 1 3 __ 4 = - 1 __

4 + w

____

+ 1 __ 4

________ + 1 __

4

2 = w

47. - 1 __ 6 + h = 1 __

6

________

+ 1 __ 6

____ + 1 __

6

h = 1 __ 3

48. -5.2 + a = -8 ________ + 5.2 _____ + 5.2 a = -2.8

49. Let a represent the amount in Luis’s acount before the deposit.

a + 500 = 4732 _______ - 500 _____ - 500 a = 4232 Luis had $4232 in his account before the deposit.

Possible answer: Luis deposited $500 and now has about $4700. So the original amount will be close to

4700 - 500 = 4200. So $4232 is a reasonable answer.

50. Solution B is incorrect. The Addition Property of Equality was used.

51. x - 10 = 12 ______ + 10 ____ + 10 x = 22

52. x - 13 = 7 ______ + 13 ____ + 13 x = 20

53. x + 8 = 16 _____ - 8 ___ - 8 x = 8

54. x - 3 = -8 _____ + 3 ___ + 3 x = -5

55. 5 + x = 6 _______ - 5 ___ - 5 x = 1

56. x - 2 = -5 _____ + 2 ___ + 2 x = -3

57. x - 4 = 9 _____ + 4 ___ + 4 x = 13

58. Let a represent the average depth of the Atlantic Ocean.

a + 30,246 = 43,126 _________ - 30,246 ________ - 30,246 a = 12,880 The average depth of the Atlantic Ocean is

12,880 ft.

Possible answer: The greatest depth is about 30,000 ft. The sum of this number and the average is about 43,000 ft. So the average will be close to

43,000 - 30,000 = 13,000. So 12,800 ft is a reasonable answer.

59. Let m represent the amount of money the band still needs.

m + 560 = 1680 _______ - 560 _____ - 560 m = 1120 Helene’s band still needs to raise $1120.

Possible answer: The band needs about $1700 and they have raised about $600. So the amount they need will be close to 1700 - 600 = 1100. So $1120 is a reasonable answer.

60. Let c represent the cost of the car. c - 1500 = 2600 ________ + 1500 ______ + 1500 c = 4100 The car cost $4100.

Possible answer: The total cost should be more than the down payment and more than the loan amount, so $4100 is a reasonable answer.

61. 63 + x = 90 ________ - 63 ____ - 63 x = 27

62. 42 + x = 90 ________ - 42 ____ - 42 x = 48

63. x + 15 = 90 ______ - 15 ____ - 15 x = 75

64a. 2000 acres; possible answer: The fire should cover twice as much area in 2 days as it does in 1, so multiply 2 by 1000 and the answer is 2000 acres.

b. 5000 acres; multiply 5 days by 1000 acres per day.

c. Divide 780 by 7.

65. Let h represent the highest score. h - 47 = 28 ______ + 47 ____ + 47 h = 75 The highest score is 75.

Possible answer: The range is about 30 and the lowest score is about 50. So the highest score will be close to 30 + 50 = 80. So 75 is a reasonable answer.

66. Possible answer: If you add 5 years to Sue’s age, you get her cousin’s age. Her cousin is 25. How old is Sue? x represents Sue’s age.

x + 5 = 25 _____ - 5 ___ - 5 x = 20 Sue is 20 years old.

67. Possible answer: greater than 10 because you will add a positive number to both sides

teSt prep

68. A: 32 is being subtracted from x so the situation must decrease by 32. Choice A has a withdrawal of 32.

69. J: Since 8 + 8 = 16, J is an equation for which a = 8 is a solution.

70a. g - 18 = 22 b. $40 g - 18 = 22 ______ + 18 ____ + 18 g = 40


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71. 3 1 __ 5 + b = 4 __

5

________

- 3 1 __ 5

_____ - 3 1 __

5

b = - 12 ___ 5

72. x - 7 __ 4 = 2 __

3

_____

+ 7 __ 4

____ + 4 __

7

x = 29 ___ 12

73. x + 7 __ 4 = 2 __

3

______

- 7 __ 4

_____ - 7 __

4

x = - 13 ___ 12

74. x - 4 __ 9 = 4 __

9

_____

+ 4 __ 9

____ + 4 __

9

x = 8 __ 9

75. p - 4 = 2 _____ + 4 ___ + 4 p = 6

5p - 20 = 5(6) - 20 = 30 - 20 = 10

76. t + 6 = 21 ____ - 6 ___ - 6 t = 15

-2t = -2(15) = -30

77. x + 3 = 15 _____ - 3 ___ - 3 x = 12

18 + 6x = 18 + 6(12) = 18 + 72 = 90

78. 2 + n = -11 _______ - 2 ____ - 2 n = -13

6n = 6(-13) = -78

solVing equAtions bY multiplYing or diViding

CheCk it OUt!

1a. p __

5 = 10

(5) ( p __

5 ) = (5)(10)

p = 50

b. -13 = y __

3

(3)(-13) = (3) ( y __

3 )

-39 = y

c. c __ 8 = 7

(8) ( c __ 8 ) = (8)(7)

c = 56

2a. 16 = 4c

16 ___ 4 = 4c ___

4

4 = c

b. 0.5y = -10

0.5y

____ 0.5

= -10 ____ 0.5

y = -20

c. 15k = 75

15k ____ 15

= 75 ___ 15

k = 5

3a. - 1 __ 4 = 1 __

5 b

(5) (- 1 __ 4 ) = (5) ( 1 __

5 b)

- 5 __ 4 = b

b. 4j

__ 6 = 2 __

3

( 6 __ 4 ) 4 __

6 j = ( 6 __

4 ) 2 __

3

j = 1

c. 1 __ 6 w = 102

(6) ( 1 __ 6 w) = (6)(102)

w = 612

4. d __ 3 = h

45 ___ 3 = h

15 = h The plane was 15,000 ft high when it began its

descent.

think and disCUss

1. Possible answer: All four properties tell you that if you perform the same operation on both sides of the equation, the equality will still hold.

2. Properties of Equality

×

6x = -12

÷

x = 40

= 4 x _ 10

x = -2


1. k __ 4 = 8

(4) ( k __ 4 ) = (4)(8)

k = 32

2. z __ 3

= -9

(3) ( z __ 3

) = (3)(-9)

z = -27

3. -2 = w ___ -7

(-7)(-2) = (-7) ( w ___ -7

)

14 = w

4. 6 = t ___ -5

(-5)(6) = (-5) ( t ___ -5

)

-30 = t

5. g ___

1.9 = 10

(1.9) ( g ___

1.9 ) = (1.9)(10)

g = 19

6. 2.4 = b __ 5

(5)(2.4) = (5) ( b __ 5

)

12 = b

7. 4x = 28

4x ___ 4 = 28 ___

4

x = 7

8. -64 = 8c

-64 ____ 8

= 8c ___ 8

-8 = c

9. -9j = -45

-9j

___ -9

= -45 ____ -9

j = 5

10. 84 = -12a

84 ____ -12

= -12a _____ -12

-7 = a

11. 4m = 10

4m ___ 4 = 10 ___

4

m = 2.5

12. 2.8 = -2h

2.8 ___ -2

= -2h ____ -2

-1.4 = h

13. 1 __ 2 d = 7

(2) ( 1 __ 2 d) = (2)(7)

d = 14

14. 15 = 5 __ 6

f

( 6 __ 5

) (15) = ( 6 __ 5

) ( 5 __ 6

f)

18 = f


1-3

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15. 2 __ 3 s = -6

( 3 __ 2 ) ( 2 __

3 s) = ( 3 __

2 ) (-6)

s = -9

16. 9 = - 3 __ 8 r

(- 8 __ 3 ) (9) = (- 8 __

3 ) (- 3 __

8 r)

-24 = r

17. 1 ___ 10

= ( 4 __ 5 ) y

( 5 __ 4 ) 1 ___

10 = ( 5 __

4 ) 4 __

5 y

1 __ 8 = y

18. 1 __ 4 v = - 3 __

4

(4) ( 1 __ 4 v) = (4) (- 3 __

4 )

v = -3

19. Let c represent the cost per child. 16c = 192

16c ____ 16

= 192 ____ 16

c = 12 The cost per child is $12.

20. Let a represent the amount of vitamin C in an apple. 80 = 10a

80 ___ 10

= 10a ____ 10

8 = a There are 8 mg of vitamin C in an apple.


21. x __ 2 = 12

(2) ( x __ 2 ) = (2)(12)

x = 24

22. -40 = b __ 5

(5)(-40) = (5) ( b __ 5 )

-200 = b

23. - j __

6 = 6

(-6) (- j __

6 ) = (-6)(6)

j = -36

24. - n __ 3 = -4

(-3) (- n __ 3 ) = (-3)(-4)

n = 12

25. - q __

5 = 30

(-5) (- q

__ 5 ) = (-5)(30)

q = -150

26. 1.6 = d __ 3

(3)(1.6) = (3) ( d __ 3 )

4.8 = d

27. v ___ 10

= 5.5

(10) ( v ___ 10

) = (10)(5.5)

v = 55

28. h ___ 8.1

= -4

(8.1) ( h ___ 8.1

) = (8.1)(-4)

h = -32.4

29. 5t = -15

5t __ 5 = -15 ____

5

t = -3

30. 49 = 7c

49 ___ 7 = 7c ___

7

7 = c

31. -12 = -12u

-12 ____ -12

= -12u _____ -12

1 = u

32. -7m = 63

-7m _____ -7

= 63 ___ -7

m = -9

33. -52 = -4c

-52 ____ -4

= -4c ____ -4

13 = c

34. 11 = -2z

11 ___ -2

= -2z ____ -2

-5.5 = z

35. 5f = 1.5

5f __ 5 = 1.5 ___

5

f = 0.3

36. -8.4 = -4n

-8.4 _____ -4

= -4n ____ -4

2.1 = n

37. 5 __ 2 k = 5

( 2 __ 5 ) ( 5 __

2 k) = ( 2 __

5 ) (5)

k = 2

38. -9 = 3 __ 4 d

( 4 __ 3 ) (-9) = ( 4 __

3 ) ( 3 __

4 ) d

-12 = d

39. - 5 __ 8 b = 10

(- 8 __ 5 ) (- 5 __

8 b) = (- 8 __

5 ) (10)

b = -16

40. - 4 __ 5 g = -12

(- 5 __ 4 ) (- 4 __

5 g) = (- 5 __

4 ) (-12)

g = 15

41. 4 __ 7 t = -2

( 7 __ 4 ) ( 4 __

7 t) = ( 7 __

4 ) (-2)

t = -3.5

42. - 4 __ 5 p = 2 __

3

(- 5 __ 4 ) (- 4 __

5 p) = (- 5 __

4 ) ( 2 __

3 )

p = - 5 __ 6

43. 2 __ 3 = - 1 __

3 q

(-3) ( 2 __ 3 ) = (-3) (- 1 __

3 q)

-2 = q

44. - 5 __ 8 = - 3 __

4 a

(- 4 __ 3 ) (- 5 __

8 ) = (- 4 __

3 ) (- 3 __

4 a)

5 __ 6 = a

45. Let s represent Alexandra’s salary before taxes.

7 ___ 10

s = 392

( 10 ___ 7 ) ( 7 ___

10 s) = ( 10 ___

7 ) (392)

s = 560 Alexandra’s salary before taxes is $560.

46. Let w represent the person’s weight on Earth.

1 __ 6 w = 16

(6) ( 1 __ 6 w) = (6)(16)

w = 96 The person weighs 96 lb on Earth.

Possible answer: The person’s weight on the Moon is about 15 lb. The answer will be close to 15 · 6 = 90. So 96 lb is reasonable.

47. Possible answer: The student divided both sides by 3 instead of multiplying both sides by 3.

x __ 3 = 15

(3) ( x __ 3 ) = (3)(15)

x = 45


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48. 4s = 36

4s ___ 4 = 36 ___

4

s = 9 in.

49. 4s = 84

4s ___ 4 = 84 ___

4

s = 21 in.

50. 4s = 100

4s ___ 4 = 100 ____

4

s = 25 yd

51. 4s = 16.4

4s ___ 4 = 16.4 ____

4

s = 4.1 cm

52. 5x = 45

5x ___ 5 = 45 ___

5

x = 9

53. -3x = 12

-3x ____ -3

= 12 ___ -3

x = -4

54. x __ 4 = 10

(4) ( x __ 4 ) = (4)(10)

x = 40

55. x __ 3 = -8

(3) ( x __ 3 ) = (3)(-8)

x = -24

56. Let x represent the total measure of all students’ heights.

x ___ 18

= 60

(18) ( x ___ 18

) = (18)(60)

x = 1080 The total measure of all students’ heights is 1080 in.

57. Let h represent the number of hours Lisa worked each week.

6.25h = 50

6.25h _____ 6.25

= 50 ____ 6.25

h = 8 Lisa worked 8 hours each week.

58. Greater than 4; you are multiplying 4 by a positive number greater than 1.

59. Let m represent the number of minutes Dion was charged for.

0.05m = 13.80

0.05m ______ 0.05

= 13.80 _____ 0.05

m = 276 Dion was charged for 276 minutes.

Possible answer: Dion’s calls cost $0.05/min. If he talked for 300 min, the cost would be 300(0.05) = $15. This is close to the cost given in

the problem, $13.80. So 276 min is reasonable.

60. Let c represent the amount of caffeine in a 12 oz caffeinated soft drink.

5c = 184

5c ___ 5 = 184 ____

5

c ≈ 37 There are about 37 mg of caffeine in a 12 oz

caffeinated soft drink.

Possible answer: The amount of caffeine in the soft

drink should be about 200 ____ 5 = 40 mg. So 37 mg is

reasonable.

61. 8y = 4x 8y = 4(-4) 8y = -16

8y

___ 8 = -16 ____

8

y = -2

62. 8y = 4x 8y = 4(-2) 8y = -8

8y

___ 8 = -8 ___

8

y = -1

63. 8y = 4x 8y = 4(0) 8y = 0

8y

___ 8 = 0 __

8

y = 0

64. 8y = 4x 8y = 4(2) 8y = 8

8y

___ 8

= 8 __ 8

y = 1

65a. number of data values

b. mean

c. mean = sum of data values __________________ number of data values

96.21 = x _____ 1926

(1926)(96.21) = (1926) ( x _____ 1926

)

185,300 acres ≈ x Possible answer: There were about 1900 fires

and the mean acreage burned was about 100. So the total acres burned will be close to 1900(100) = 190,000. So 185,300 is reasonable.

66. m __ 6 = 1

(6) ( m __ 6 ) = (6)(1)

m = 6

67. 4x = 28

4x ___ 4

= 28 ___ 4

x = 7

68. 1.2h = 14.4

1.2h ____ 1.2

= 14.4 ____ 1.2

h = 12

69. 1 __ 5

x = 121

(5) ( 1 __ 5

x) = (5)(121)

x = 605

70. 2w = 26

2w ___ 2 = 26 ___

2

w = 13

71. 4b = 3 __ 4

( 1 __ 4 ) (4b) = ( 1 __

4 ) ( 3 __

4 )

b = 3 ___ 16

72. 5y = 11

5y

___ 5 = 11 ___

5

y = 11 ___ 5

73. n ___ 1.9

= 3

(1.9) ( n ___ 1.9

) = (1.9)(3)

n = 5.7

74. Let m represent the mean weight of an adult male mouse.

16m = 480

16m ____ 16

= 480 ____ 16

m = 30 The mean weight of an adult male mouse is 30 g.

Possible answer: An adult male rat weighs about 500 g, and this is about 20 times the weight of an adult mouse. So the mouse will weigh

close to 500 ____ 20

= 25 g. So 30 g is reasonable.


CS10_A1_MESK710372_C01.indd 7 3/30/11 10:30:48 PM

75. Let g represent the average weight of a gerbil at birth.

2 __ 3 g = 2

( 3 __ 2 ) ( 2 __

3 g) = ( 3 __

2 ) (2)

g = 3 The average weight of a gerbil at birth is 3 g.

Possible answer: A hamster weighs 2 g at birth, so the gerbil’s weight will be greater than 2. So 3 g is reasonable.

76. Possible answer: Ryan has 3 times as many baseball cards as David. Ryan has 42 cards. How many cards does David have?

3x = 42

3x ___ 3 = 42 ___

3

x = 14 This represents the number of cards that David has.

teSt prep

77. D; Since situation D says “Mattie had 2 more,” the situation involves addition. Since the given equation does not contain addition, D cannot represent the equation.

78. J; Since x is multiplied by 3, the first step is to divide both sides by 3. Therefore, equation J is correct.

79. B; There are 540° in a pentagon and 5 angles. If x is the size of each angle, then 5 angles must add to 540°. Therefore 5x = 540 is correct.

80. H; Since 10 ___ 2 = 5, m = 10 is a solution for equation

H.

81a. 6c = 4.80; 6 is the number of cans of cat food; c is the cost per can; $4.80 is the total cost.

b. $0.80 6c = 4.80

6c ___ 6 = 4.80 ____

6

c = 0.80One can of cat food costs $0.80.


82. (3 1 __ 5 ) b = 4 __

5

16 ___ 5 b = 4 __

5

( 5 ___ 16

) 16 ___ 5 b = ( 5 ___

16 ) 4 __

5

b = 1 __ 4

83. (1 1 __ 3 ) x = 2 2 __

3

4 __ 3 x = 8 __

3

( 3 __ 4 ) 4 __

3 x = ( 3 __

4 ) 8 __

3

x = 2

84. (5 4 __ 5 ) x = -52 1 __

5

29 ___ 5 x = - 261 ____

5

( 5 ___ 29

) 29 ___ 5 x = ( 5 ___

29 ) (- 261 ____

5 )

x = -9

85. (-2 9 ___ 10

) k = -26 1 ___ 10

-29 ____ 10

k = - 261 ____ 10

(- 10 ___ 29

) (- 29 ___ 10

k) = (- 10 ___ 29

) (- 261 ____ 10

)

k = 9

86. (1 2 __ 3 ) w = 15 1 __

3

5 __ 3 w = 46 ___

3

( 3 __ 5 ) 5 __

3 w = ( 3 __

5 ) 46 ___

3

w = 46 ___ 5

87. (2 1 __ 4 ) d = 4 1 __

2

9 __ 4 d = 9 __

2

( 4 __ 9 ) 9 __

4 d = ( 4 __

9 ) 9 __

2

d = 2

88. 2p = 4

2p

___ 2 = 4 __

2

p = 2

6p + 10 = 6(2) + 10 = 12 + 10 = 22

89. 6t = 24

6t __ 6 = 24 ___

6

t = 4

-5t = -5(4) = -20

90. 3x = 15

3x ___ 3 = 15 ___

3

x = 5

12 - 4x = 12 - 4(5) = 12 - 20 = -8

91. n __ 2 = -11

(2) ( n __ 2 ) = 2(-11)

n = -22

6n = 6(-22) = -132

92. a 93. Multiply both sides by a.

94a. d = rt

400 = 25t

400 ____ 25

= 25t ___ 25

16 = t

b. d = rt

400 = 50t

400 ____ 50

= 50t ___ 50

8 = t

c. t is divided in half (from 16 to 8).

d. t is divided in half.

solVing two-step And multi-step equAtions

CheCk it OUt!

1a. -4 + 7x = 3 ________ + 4 ___ + 4 7x = 7

7x ___ 7 = 7 __

7

x = 1

b. 1.5 = 1.2y - 5.7 _____ + 5.7 _________ + 5.7 7.2 = 1.2y

7.2 ___ 1.2

= 1.2y

____ 1.2

6 = y


1-4

CS10_A1_MESK710372_C01.indd 8 3/30/11 10:30:49 PM

c. n __ 7 + 2 = 2

_____ - 2 ___ - 2

n __ 7 = 0

(7) ( n __ 7 ) = (7)(0)

n = 0

2a. 2x ___ 5 - 1 __

2 = 5

______

+ 1 __ 2

____ + 1 __

2

2x ___ 5 = 11 ___

2

( 5 __ 2 ) 2 __

5 x = ( 5 __

2 ) 11 ___

2

x = 55 ___ 4

b. 3 __ 4 u + 1 __

2 = 7 __

8

_______

- 1 __ 2

____ - 1 __

2

3 __ 4 u = 3 __

8

( 4 __ 3 ) 3 __

4 u = ( 4 __

3 ) 3 __

8

u = 1 __ 2

c. 1 __ 5 n - 1 __

3 = 8 __

3

______

+ 1 __ 3

____ + 1 __

3

1 __ 5 n = 3

(5) ( 1 __ 5 n) = (5)(3)

n = 15

3a. 2a + 3 - 8a = 8 2a - 8a + 3 = 8 -6a + 3 = 8 ________ - 3 ___ - 3 -6a = 5

-6a ____ -6

= 5 ___ -6

a = - 5 __ 6

b. -2(3 - d) = 4 (-2)(3) + (-2)(-d) = 4 -6 + 2d = 4 ________ + 6 ___ + 6 2d = 10

2d ___ 2 = 10 ___

2

d = 5

c. 4(x - 2) + 2x = 40 (4)(x) + (4)(-2) + 2x = 40 4x - 8 + 2x = 40 4x + 2x - 8 = 40 6x - 8 = 40 ______ + 8 ___ + 8 6x = 48

6x ___ 6 = 48 ___

6

x = 8

4. Let x represent Sara’s monthly fee. 12x + 15.95 = 735.95 __________ - 15.95 _______ - 15.95 12x = 720.00

12x ____ 12

= 720.00 ______ 12

x = 60.00 Sara’s monthly fee was $60.00.

5. 2x + 4 = -24 _____ - 4 ____ - 4 2x = -28

2x ___ 2 = -28 ____

2

x = -14

3x = 3(-14) = -42

think and disCUss

1. Possible answer: To solve 2x + 1 = 7, first subtract 1 from both sides and then divide both sides by 2. To solve 2x - 1 = 7, first add 1 to both sides and then divide both sides by 2.

2.

2x + 1 = 9 - 2 = 1

x = 4 x = 9

2x = 8

x _ 3

= 3 x _ 3

Solving Multi-Step Equations


1. 4a + 3 = 11 ______ - 3 ___ - 3 4a = 8

4a ___ 4 = 8 __

4

a = 2

2. 8 = 3r - 1 ___ + 1 ______ + 1 9 = 3r

9 __ 3

= 3r __ 3

3 = r

3. 42 = -2d + 6 ___ - 6 _______ - 6 36 = -2d

36 ___ -2

= -2d ____ -2

-18 = d

4. x + 0.3 = 3.3 ______ - 0.3 _____ - 0.3 x = 3

5. 15y + 31 = 61 _______ - 31 ____ - 31 15y = 30

15y

____ 15

= 30 ___ 15

y = 2

6. 9 - c = -13 - 9 - 9 -c = -22 (-1)(-c) = (-1)(-22) c = 22

7. x __ 6 + 4 = 15

_____ - 4 ___ - 4

x __ 6 = 11

6 ( x __ 6 ) = 6(11)

x = 66

8. 1 __ 3

y + 1 __ 4

= 5 ___ 12

______

- 1 __ 4

____

- 1 __ 4

1 __ 3

y = 1 __ 6

3 ( 1 __ 3

y) = 3 ( 1 __ 6

)

y = 1 __ 2


CS10_A1_MESK710372_C01.indd 9 3/30/11 10:30:51 PM

9. 2 __ 7 j - 1 __

7 = 3 ___

14

______

+ 1 __ 7

____ + 1 __

7

2 __ 7 j = 5 ___

14

( 7 __ 2 ) 2 __

7 j = ( 7 __

2 ) 5 ___

14

j = 5 __ 4

10. 15 = a __ 3 - 2

___ + 2 _____ + 2

17 = a __ 3

3(17) = 3 ( a __ 3 )

51 = a

11. 4 - m __ 2 = 10

_______ -4 ___ - 4

- m __ 2 = 6

(-2) (- m __ 2 ) = (-2)(6)

m = -12

12. x __ 8 - 1 __

2 = 6

_____

+ 1 __ 2

____ + 1 __

2

x __ 8 = 13 ___

2

8 ( x __ 8 ) = 8 ( 13 ___

2 )

x = 52

13. 28 = 8x + 12 - 7x 28 = 8x - 7x + 12 28 = x + 12 ____ - 12 ______ - 12 16 = x

14. 2y - 7 + 5y = 0 2y + 5y - 7 = 0 7y - 7 = 0 ______ + 7 ___ + 7 7y = 7

7y

___ 7 = 7 __

7

y = 1

15. 2.4 = 3(m + 4) 2.4 = (3)(m) + (3)

(4) 2.4 = 3m + 12 ______ - 12 _______ - 12 -9.6 = 3m

-9.6 _____ 3 = 3m ___

3

-3.2 = m

16. 3(x - 4) = 48 (3)(x) + (3)(-4) = 48 3x - 12 = 48 _______ + 12 ____ + 12 3x = 60

3x ___ 3 = 60 ___

3

x = 20

17. 4t + 7 - t = 19 4t - t + 7 = 19 3t + 7 = 19 _____ - 7 ___ - 7 3t = 12

3t __ 3 = 12 ___

3

t = 4

18. 5(1 - 2w) + 8w = 15 (5)(1) + (5)(-2w) + 8w = 15 5 - 10w + 8w = 15 5 - 2w = 15 ________ - 5 ___ - 5 -2w = 10

-2w ____ -2

= 10 ___ -2

w = -5

19. Let x represent the number of daily bus passes Paul bought.

1.50x + 7 = 29.50 ________ - 7 ______ - 7 1.50x = 22.50

1.50x _____ 1.50

= 22.50 _____ 1.50

x = 15

Paul bought 15 daily bus passes.

20. 3x - 13 = 8 _______ + 13 ____ + 13 3x = 21

3x ___ 3 = 21 ___

3

x = 7

x - 4 = 7 - 4 = 3

21. 3(x + 1) = 7 (3)(x) + (3)(1) = 7 3x + 3 = 7 ______ - 3 ___ - 3 3x = 4

22. -3(y - 1) = 9

(-3)(y) + (-3)(-1) = 9 -3y + 3 = 9 _______ - 3 ___ - 3 -3y = 6

-3y

____ -3

= 6 ___ -3

y = -2

1 __ 2 y = 1 __

2 (-2)

= -1

23. 4 - 7x = 39 ________ - 4 ___ - 4 -7x = 35

-7x ____ -7

= 35 ___ -7

x = -5

x + 1 = -5 + 1 = -4


24. 5 = 2g + 1 ___ - 1 ______ - 1 4 = 2g

4 __ 2 =

2g ___

2

2 = g

25. 6h - 7 = 17 _____ + 7 ___ + 7 6h = 24

6h ___ 6 = 24 ___

6

h = 4

26. 0.6v + 2.1 = 4.5 ________ - 2.1 _____ - 2.1 0.6v = 2.4

0.6v ____ 0.6

= 2.4 ___ 0.6

v = 4

27. 3x + 3 = 18 _____ - 3 ___ - 3 3x = 15

3x ___ 3 = 15 ___

3

x = 5

28. 0.6g + 11 = 5 ________ - 11 ____ - 11 0.6g = -6

0.6g

____ 0.6

= -6 ___ 0.6

g = -10

29. 32 = 5 - 3t ___ - 5 _______ - 5 27 = -3t

27 ___ -3

= -3t ____ -3

-9 = t


CS10_A1_MESK710372_C01.indd 10 3/30/11 10:30:53 PM

30. 2d + 1 __ 5 = 3 __

5

______

- 1 __ 5

____ - 1 __

5

2d = 2 __ 5

1 __ 2 (2d) = 1 __

2 ( 2 __

5 )

d = 1 __ 5

31. 1 = 2x + 1 __ 2

- 1 __ 2

______ - 1 __

2

1 __ 2 = 2x

1 __ 2 ( 1 __

2 ) = 1 __

2 (2x)

1 __ 4 = x

32. z __ 2 + 1 = 3 __

2

_____ - 1 ___ - 1

z __ 2 = 1 __

2

2 ( z __ 2 ) = 2 ( 1 __

2 )

z = 1

33. 2 __ 3 =

4j __

6

( 6 __ 4 ) 2 __

3 = ( 6 __

4 ) 4 __

6 j

1 = j

34. 3 __ 4 = 3 __

8 x - 3 __

2

____

+ 2 __ 3

_______ + 3 __

2

9 __ 4 = 3 __

8 x

( 8 __ 3 ) 9 __

4 = ( 8 __

3 ) 3 __

8 x

6 = x

35. 1 __ 5 - x __

5 = - 2 __

5

________

- 1 __ 5

____ - 1 __

5

- x __ 5 = - 3 __

5

(-5) (- x __ 5 ) = (-5) (- 3 __

5 )

x = 3

36. 6 = -2(7 - c) 6 = (-2)(7) + (-2)(-c) 6 = -14 + 2c ____ + 14 _________ + 14 20 = 2c

20 ___ 2 = 2c ___

2

10 = c

37. 5(h - 4) = 8 (5)(h) + (5)(-4) = 8 5h - 20 = 8 _______ + 20 ____ + 20 5h = 28

5h ___ 5 = 28 ___

5

h = 28 ___ 5

38. -3x - 8 + 4x = 17 -3x + 4x - 8 = 17 x - 8 = 17 _____ + 8 ___ + 8 x = 25

39. 4x + 6x = 30 10x = 30

10x ____ 10

= 30 ___ 10

x = 3

40. 2(x + 3) = 10 (2)(x) + (2)(3) = 10 2x + 6 = 10 ______ - 6 ___ -6 2x = 4

2x ___ 2 = 4 __

2

x = 2

41. 17 = 3(p - 5) + 8 17 = (3)(p) + (3)(-5) + 8 17 = 3p - 15 + 8 17 = 3p - 7 ___ +7 ______ + 7 24 = 3p

24 ___ 3 =

3p ___

3

8 = p

42. Let w represent the number of weeks. 245 = 125 + 15w _____ - 125 ___________ - 125 120 = 15w

120 ____ 15

= 15w ____ 15

8 = w Jennifer needs to save for another 8 weeks to buy

the bike.

43. 2x + 13 = 17 ______ - 13 ____ - 13 2x = 4

2x ___ 2 = 4 __

2

x = 2

3x + 1 = 3(2) + 1 = 6 + 1 = 7

44. -(x - 1) = 5 (-1)(x)+(-1)(-1) = 5 -x + 1 = 5 ______ - 1 ___ - 1 -x = 4 (-1)(-x) = (-1)(4) x = -4 -4x = -4(-4) = 16

45. 5(y+10) = 40 (5)(y) + (5)(10) = 40 5y + 50 = 40 _______ - 50 ____ - 50 5y = -10

5y

___ 5 = -10 ____

5

y = -2

1 __ 4 y

= 1 __ 4 (-2)

= - 1 __ 2

46. 9 - 6x = 45 ________ - 9 ___ - 9 -6x = 36

-6x ____ -6

= 36 ___ -6

x = -6

x - 4 = -6 - 4 = -10

47. (2x + 7) + 30 + 63 = 180 2x + 100 = 180 ________ - 100 _____ - 100 2x = 80

2x ___ 2 = 80 ___

2

x = 40


CS10_A1_MESK710372_C01.indd 11 3/30/11 10:30:55 PM

48. x + x + 115 = 180 2x + 115 = 180 ________ - 115 _____ - 115 2x = 65

2x ___ 2 = 65 ___

2

x = 32.5

49. (4x - 80) + 60 + 60 = 180 4x + 40 = 180 _______ - 40 - 40 4x = 140

4x ___ 4 = 140 ____

4

x = 35

50. 2n - 7 = 19 ______ + 7 ___ + 7 2n = 26

2n ___ 2 = 26 ___

2

n = 13

51. 8 - 3n = 2 - 8 - 8 -3n = -6

-3n ____ -3

= -6 ___ -3

n = 2

52. 2n + 5 = 11 ______ - 5 ___ - 5 2n = 6

2n ___ 2 = 6 __

2

n = 3

53a. Let s represent the number of years in a score. 1963 - 5s = 1863 ___________ - 1963 ______ - 1963 -5s = -100

-5s ____ -5

= -100 _____ -5

s = 20 There are 20 years in a score.

b. Let n repesent the number of scores. ns = 60 n(20) = 60

20n ____ 20

= 60 ___ 20

n = 3 There are 3 scores in 60 years.

54. 3t + 44 = 50 ______ - 44 ____ - 44 3t = 6

3t __ 3 = 6 __

3

t = 2

55. 3(x - 2) = 18 (3)(x) + (3)(-2) = 18 3x - 6 = 18 ______ + 6 ___ + 6 3x = 24

3x ___ 3 = 24 ___

3

x = 8

56. 15 = c __ 3 - 2

___ + 2 _____ + 2

17 = c __ 3

3(17) = 3 ( c __ 3 )

51 = c

57. 2x + 6.5 = 15.5 _______ - 6.5 _____ - 6.5 2x = 9

2x ___ 2 = 9 __

2

x = 4.5

58. 3.9w - 17.9 = -2.3 __________ + 17.9 ______ + 17.9 3.9w = 15.6

3.9w ____ 3.9

= 15.6 ____ 3.9

w = 4

59. 17 = x - 3(x + 1) 17 = x + (-3)(x) + (-3)(1) 17 = x - 3x - 3 17 = -2x - 3 ___ + 3 _______ + 3 20 = -2x

20 ___ -2

= -2x ____ -2

-10 = x

60. 5x + 9 = 39 _____ - 9 ___ - 9 5x = 30

5x ___ 5 = 30 ___

5

x = 6

61. 15 + 5.5m = 70 ___________ - 15 ____ - 15 5.5m = 55

5.5m _____ 5.5

= 55 ___ 5.5

m = 10

62. Let k represent the height of a kiwi. 4k + 20 = 108 ______ - 20 ____ - 20 4k = 88

4k ___ 4 = 88 ___

4

k = 22 A kiwi is 22 in. tall.

Possible answer: The height of an ostrich will be more than 4 times the height of a kiwi. 108 is more than 4(22) = 88, so 22 in. is reasonable.

63. Let k represent the height of a kakapo. 5k - 70 = 60 ______ + 70 ____ + 70 5k = 130

5k ___ 5 = 130 ____

5

k = 26 A kakapo is 26 in. tall.

Possible answer: The height of a kakapo will be more than 1 __

5 the height of an emu. 26 is more

than 1 __ 5 (60) = 12, so it is a reasonable answer.

64. Let n represent the first number. (n) + (n + 1) = 57 2n + 1 = 57 ______ - 1 ___ - 1 2n = 56

2n ___ 2 = 56 ___

2

n = 28 The numbers are n = 28 and n + 1 = 29.


CS10_A1_MESK710372_C01.indd 12 3/30/11 10:30:57 PM

65. Let n represent Stan’s age. (n) + (n + 1) + (n + 2) = 111 3n + 3 = 111 ______ - 3 ___ - 3 3n = 108

3n ___ 3 = 108 ____

3

n = 36 Stan is n = 36, Mark is n + 1 = 37, and Wayne is n + 2 = 38.

66. Let n represent the first number. (n) + (n + 2) = 206 2n + 2 = 206 ______ - 2 ___ - 2 2n = 204

2n ___ 2 = 204 ____

2

n = 102 The numbers are n = 102 and n + 2 =104.

67a. Cost of Fighting Fire

Acres Cost ($)

100 22,500

200 45,000

500 112,500

1000 225,000

1500 337,500

n 225n

b. c = 225n

68. Possible answer: Distribute 2 or subtract 3 from both sides.

69. Possible answer: Simplify both sides if necessary. Find the term containing the variable. Undo any addition or subtraction on this term by using inverse operations. Then undo any multiplication or division on this term by using inverse operations.

teSt prep

70. D; g represents the number of shirts Greg sold. Lin sold g + 4 and Fran sold 3(g + 4) = 3g + 12. Therefore, the total number of shirts sold is

g + g + 4 + 3g + 12 = 16 + 5g, so D is correct.

71. H; Solving for m reveals m = 6, so 7m - 5 is 42 - 5 = 37.

72. C; Since m is multiplied by 0.06, there is a cost of $0.06 for each mile. Since 48 is added to 0.06m, there is an additional fee of $48. Therefore C is correct.

73. 27 sales 150 + 2s = 204 __________ - 150 _____ - 150 2s = 54

2s ___ 2 = 54 ___

2

s = 27


74. 9 __ 2 x + 18 + 3x = 11 ___

2

9 __ 2 x + 3x + 18 = 11 ___

2

15 ___ 2 x + 18 = 11 ___

2

________ - 18 ____ - 18

15 ___ 2 x = - 25 ___

2

( 2 ___ 15

) ( 15 ___ 2 x) = ( 2 ___

15 ) (- 25 ___

2 )

x = - 5 __ 3

75. 15 ___ 4 x - 15 = 33 ___

4

________ + 15 ____ + 15

15 ___ 4 x = 93 ___

4

( 4 ___ 15

) 15 ___ 4 x = ( 4 ___

15 ) 93 ___

4

x = 6 1 __ 5

76. (x + 6) - (2x + 7) - 3x = -9 x + 6 + (-1)(2x) + (-1)(7) - 3x = -9 x + 6 -2x -7 - 3x = -9 x - 2x - 3x + 6 - 7 = -9 -4x - 1 = -9 _______ + 1 ___ + 1 -4x = -8

-4x ____ -4

= -8 ___ -4

x = 2

77. (4x + 2) - (12x + 8) + 2(5x - 3) = 6 + 11 4x + 2 - 12x - 8 + 10x - 6 = 6 + 11 4x - 12x + 10x + 2 - 8 - 6 = 6 + 11 2x - 12 = 17 _______ + 12 ____ + 12 2x = 29

2x ___ 2

= 29 ___ 2

x = 14.5

78. 4x + 3b = -1 4(2) + 3b = -1 8 + 3b = -1 ________ - 8 ___ - 8 3b = -9

3b ___ 3 = -9 ___

3

b = -3

79. 2x - 3b = 0 2(-9) - 3b = 0 -18 - 3b = 0 ________ +18 ____ + 18 -3b = 18

-3b ____ -3

= 18 ___ -3

b = -6

80a. p = nc - e 2500 = 2000c - 800 _____ + 800 ___________ + 800 3300 = 2000c

3300 _____ 2000

= 2000c ______ 2000

1.65 = c

b. p = nc - e 2500 = 1000c - 800 _____ + 800 ___________ + 800 3300 = 1000c

3300 _____ 1000

= 1000c ______ 1000

3.3 = c

c. c doubles


CS10_A1_MESK710372_C01.indd 13 3/30/11 10:30:58 PM

solVing equAtions with VAriAbles on both sides

CheCk it OUt!

1a. 4b + 2 = 3b ________ - 3b ____ - 3b b + 2 = 0 _____ - 2 ___ - 2 b = -2

b. 0.5 + 0.3y = 0.7y - 0.3 _________ - 0.3y ___________ - 0.3y 0.5 = 0.4y - 0.3 _____ + 0.3 _________ + 0.3 0.8 = 0.4y

0.8 ___ 0.4

= 0.4y

____ 0.4

2 = y

2a. 1 __ 2 (b + 6) = 3 __

2 b - 1

1 __ 2 (b) + 1 __

2 (6) = 3 __

2 b - 1

1 __ 2 b + 3 = 3 __

2 b - 1

________

- 1 __ 2 b

________ - 1 __

2 b

3 = b - 1 ___ + 1 _____ + 1 4 = b

b. 3x + 15 - 9 = 2(x + 2) 3x + 15 - 9 = 2(x) + 2(2) 3x + 15 - 9 = 2x + 4 3x + 6 = 2x + 4 ________ - 2x ________ - 2x x + 6 = 4 _____ - 6 ___ - 6 x = -2

3a. 4y + 7 - y = 10 + 3y 7 + 3y = 10 + 3y ______ - 3y _______ - 3y 7 = 10 7 no solution

b. 2c + 7 + c = -14 + 3c + 21 3c + 7 = 3c + 7 ________ - 3c ________ - 3c 7 = 7 3 all real numbers

4. Let g represent Greg’s age. 4g - 3 = 3g + 7 ________ - 3g ________ - 3g g - 3 = 7 _____ + 3 ___ + 3 g = 10 Greg is 10 years old.

think and disCUss

1. Equation b, 8.3x - 9 + 0.7x = 2 + 9x - 11 is an identity. The equation simplifies to 9x - 9 = 9x - 9 which is true for all values of x.

2. An equation with variables on both sides can have…

One solution: 5x - 4 = 4x + 5

Many solutions: 5x - 4 = 5x - 4

No solution: 5x - 4 = 5x - 3


1. Possible answer: After simplifying, the expressions on either side of the equal sign are the same.

2. 2c - 5 = c + 4 _______ - c _______ - c c - 5 = 4 _____ + 5 ___ + 5 c = 9

3. 8r + 4 = 10 + 2r ________ - 2r ______ - 2r 6r + 4 = 10 _____ - 4 ___ - 4 6r = 6

6r __ 6 = 6 __

6

r = 1

4. 2x - 1 = x + 11 _______ - x ________ - x x - 1 = 11 _____ + 1 ___ + 1 x = 12

5. 28 - 0.3y = 0.7y - 12 ________ + 0.3y __________ + 0.3y 28 = y - 12 ____ + 12 ______ + 12 40 = y

6. -2(x + 3) = 4x - 3 -2(x) - 2(3) = 4x - 3 -2x - 6 = 4x - 3 ________ + 2x ________ + 2x -6 = 6x - 3 ___ + 3 ______ + 3 -3 = 6x

-3 ___ 6 = 6x ___

6

- 1 __ 2 = x

7. 3c - 4c + 1 = 5c + 2 + 3 -c + 1 = 5c + 5 _______ + c _______ + c 1 = 6c + 5 ___ - 5 ______ - 5 -4 = 6c

-4 ___ 6 = 6c ___

6

- 2 __ 3 = c

8. 5 + 3(q - 4) = 2(q + 1) 5 + 3(q) + 3(-4) = 2(q) + 2(1) 5 + 3q -12 = 2q + 2 3q - 7 = 2q + 2 ________ - 2q ________ - 2q q - 7 = 2 _____ + 7 ___ + 7 q = 9


1-5

CS10_A1_MESK710372_C01.indd 14 3/30/11 10:31:00 PM

9. 5 - (t + 3) = -1 + 2(t - 3) 5 - (t) - (3) = -1 + 2(t) + 2(-3) 5 - t - 3 = -1 + 2t - 6 2 - t = 2t - 7 ____ + t ______ + t 2 = 3t - 7 ___ + 7 _____ + 7 9 = 3t

9 __ 3 = 3t __

3

3 = t

10. 7x - 4 = -2x + 1 + 9x - 5 7x - 4 = 7x - 4 ________ - 7x _______ -7x -4 = -4 3 all real numbers

11. 8x + 6 - 9x = 2 - x - 15 6 - x = -13 - x _____ + x _______ + x 6 = -13 7 no solution

12. 6y = 8 - 9 + 6y 6y = -1 + 6y ____ - 6y _______ - 6y 0 = -1 7 no solution

13. 6 - 2x - 1 = 4x + 8 - 6x - 3 5 - 2x = 5 - 2x ______ + 2x ______ + 2x 5 = 5 3 all real numbers

14a. Let x represent the number of hours. 376 + 12x = 280 + 15x _________ - 12x _________ - 12x 376 = 280 + 3x _____ - 280 ___________ - 280 96 = 3x

96 ___ 3 = 3x ___

3

32 = x The two companies cost the same amount for a

32-hour job.

b. 376 + 12x = 376 + 12(32) = 376 + 384 = $760


15. 7a - 17 = 4a + 1 _________ - 4a ________ - 4a 3a - 17 = 1 _______ + 17 ____ + 17 3a = 18

3a ___ 3 = 18 ___

3

a = 6

16. 2b - 5 = 8b + 1 ________ - 2b ________ - 2b -5 = 6b + 1 ___ - 1 ______ - 1 -6 = 6b

-6 ___ 6 = 6b ___

6

-1 = b

17. 4x - 2 = 3x + 4 ________ - 3x ________ - 3x x - 2 = 4 _____ + 2 ___ + 2 x = 6

18. 2x - 5 = 4x - 1 ________ - 2x ________ - 2x -5 = 2x - 1 ___ + 1 ______ + 1 -4 = 2x

-4 ___ 2

= 2x ___ 2

-2 = x

19. 8x - 2 = 3x + 12.25 ________ - 3x ___________ - 3x 5x - 2 = 12.25 ______ + 2 ______ + 2 5x = 14.25

5x ___ 5 = 14.25 _____

5

x = 2.85

20. 5x + 2 = 3x ________ - 5x ____ - 5x 2 = -2x

2 ___ -2

= -2x ____ -2

-1 = x

21. 3c - 5 = 2c + 5 ________ - 2c ________ - 2c c - 5 = 5 _____ + 5 ___ + 5 c = 10

22. -17 - 2x = 6 - x ________ + 2x ______ + 2x -17 = 6 + x ____ - 6 ________ - 6 -23 = x

23. 3(t - 1) = 9 + t 3(t) + 3(-1) = 9 + t 3t - 3 = 9 + t ______ - t _____ - t 2t - 3 = 9 _____ + 3 ___ + 3 2t = 12

2t __ 2

= 12 ___ 2

t = 6

24. 5 - x - 2 = 3 + 4x + 5 3 - x = 4x + 8 _____ + x _______ + x 3 = 5x + 8 ___ - 8 ______ - 8 -5 = 5x

-5 ___ 5 = 5x ___

5

-1 = x

25. 2(x + 4) = 3(x - 2) 2(x) + 2(4) = 3(x) + 3( -2) 2x + 8 = 3x - 6 ________ - 2x ________ - 2x 8 = x - 6 ___ + 6 _____ + 6 14 = x

26. 3m - 10 = 2(4m - 5) 3m - 10 = 2(4m) + 2(-5) 3m - 10 = 8m - 10 _________ - 3m _________ - 3m -10 = 5m - 10 ____ + 10 _______ + 10 0 = 5m

0 __ 5 = 5m ___

5

0 = m


CS10_A1_MESK710372_C01.indd 15 3/30/11 10:31:02 PM

27. 5 - (n - 4) = 3(n + 2) 5 - 1(n) - 1(-4) = 3(n) + 3(2) 5 - n + 4 = 3n + 6 9 - n = 3n + 6 _____ + n _______ + n 9 = 4n + 6 ___ - 6 ______ - 6 3 = 4n

3 __ 4 = 4n ___

4

3 __ 4 = n

28. 6(x + 7) - 20 = 6x 6(x) + 6(7) - 20 = 6x 6x + 42 - 20 = 6x 6x + 22 = 6x _________ - 6x - 6x 22 = 0 7 no solution

29. 8(x + 1) = 4x - 8 8(x) + 8(1) = 4x - 8 8x + 8 = 4x - 8 ________ - 4x ________ - 4x 4x + 8 = -8 ______ - 8 ____ - 8 4x = -16

4x ___ 4 = -16 ____

4

x = -4

30. x - 4 - 3x = -2x - 3 - 1 -2x - 4 = -2x - 4 ________ + 2x ________ + 2x -4 = -4 3 all real numbers

31. -2(x + 2) = -2x + 1 -2(x) - 2(2) = -2x + 1 -2x - 4 = -2x + 1 ________ + 2x ________ + 2x -4 = 1 7 no solution

32. 2(x + 4) - 5 = 2x + 3 2(x) + 2(4) - 5 = 2x + 3 2x + 8 - 5 = 2x + 3 2x + 3 = 2x + 3 ________ + 2x ________ + 2x 3 = 3 3 all real numbers

33a. Let w represent the number of weeks. 150 + 2w = 195 - w ________ + w _______ + w 150 + 3w = 195 __________ -150 _____ -150 3w = 45

3w ___ 3 = 45 ___

3

w = 15

Justin and Tyson will weigh the same amount in 15 weeks.

b. 195 - w = 195 - 15 = 180 lb

34. 3(x + 4) = 18 + x 3(x) + 3(4) = 18 + x 3x + 12 = 18 + x ________ - x ______ - x 2x + 12 = 18 _______ - 12 ____ - 12 2x = 6

2x ___ 2 = 6 __

2

x = 3

35. x - 30 = 14 - 3x _________ + 3x _______ + 3x 4x - 30 = 14 _______ + 30 ____ + 30 4x = 44

4x ___ 4 = 44 ___

4

x = 11

36. 2x - 2 = x + 64 _______ - x ________ - x x - 2 = 64 _____ + 2 ___ + 2 x = 66

37. 2x - 2 = 4x + 6 ________ - 2x ________ - 2x -2 = 2x + 6 ___ - 6 ______ - 6 -8 = 2x

-8 ___ 2 = 2x ___

2

-4 = x

38. 3x + 5 = 2x + 2 ________ - 2x ________ - 2x x + 5 = 2 _____ - 5 ___ -5 x = -3

39. 4x + 3 = 5x - 4 ________ - 4x ________ - 4x 3 = x - 4 ___ + 4 _____ + 4 7 = x

40. - 2 __ 5 p + 2 = 1 __

5 p + 11

________

+ 2 __ 5 p

_________ + 2 __

5 p

2 = 3 __ 5 p + 11

____ - 11 _______ - 11

-9 = 3 __ 5 p

( 5 __ 3 ) (-9) = ( 5 __

3 ) ( 3 __

5 p)

-15 = p

41. 5x + 24 = 2x + 15 _________ - 2x _________ - 2x 3x + 24 = 15 _______ - 24 ____ - 24 3x = -9

3x ___ 3 = -9 ___

3

x = -3

42. 5x - 10 = 14 - 3x _________ + 3x _______ + 3x 8x - 10 = 14 _______ + 10 ____ + 10 8x = 24

8x ___ 8 = 24 ___

8

x = 3

43. 12 - 6x = 10 - 5x _______ + 6x _______ + 6x 12 = 10 + x ____ - 10 _________ - 10 2 = x

44. 5x - 7 = -6x - 29 ________ + 6x _________ + 6x 11x - 7 = -29 ______ + 7 ____ + 7 11x = -22

11x ___ 11

= -22 ____ 11

x = -2

45. 1.8x + 2.8 = 2.5x + 2.1 ___________ - 1.8x ___________ - 1.8x 2.8 = 0.7x + 2.1 _____ - 2.1 _________ - 2.1 0.7 = 0.7x

0.7 ___ 0.7

= 0.7x ____ 0.7

1 = x


CS10_A1_MESK710372_C01.indd 16 3/30/11 10:31:04 PM

46. 2.6x + 18 = 2.4x + 22 __________ - 2.4x __________ - 2.4x 0.2x + 18 = 22 ________ - 18 ____ - 18 0.2x = 4

0.2x ____ 0.2

= 4 ___ 0.2

x = 20

47. 1 - 3x = 2x + 8 ______ + 3x ________ + 3x 1 = 5x + 8 ___ - 8 ______ - 8 -7 = 5x

-7 ___ 5 = 5x ___

5

- 7 __ 5 = x

48. 1 __ 2 (8 - 6h) = h

1 __ 2 (8) + 1 __

2 (-6h) = h

4 - 3h = h ______ + 3h ____ + 3h 4 = 4h

4 __ 4 = 4h ___

4

1 = h

49. 3(x + 1) = 2x + 7 3(x) + 3(1) = 2x + 7 3x + 3 = 2x + 7 _______ - 2x _______ - 2x x + 3 = 7 _____ - 3 ___ - 3 x = 4

50. 9x - 8 + 4x = 7x + 16 13x - 8 = 7x + 16 ________ - 7x _________ - 7x 6x - 8 = 16 ______ + 8 ___ + 8 6x = 24

6x ___ 6 = 24 ___

6

x = 4

51. 3(2x - 1) + 5 = 6(x + 1) 3(2x) +3(-1) + 5 = 6(x) + 6(1) 6x - 3 + 5 = 6x + 6 6x + 2 = 6x + 6 2 = 6

There is no solution.

52a. Let m represent the number of miles. 40 + 15 + 0.25m = 45 + 0.35m 55 + 0.25m = 45 + 0.35m __________ - 0.25m __________ - 0.25m 55 = 45 + 0.10m ____ - 45 ____________ - 45 10 = 0.10m

10 ____ 0.10

= 0.10m ______ 0.10

100 = m

45 + 0.35m = 45 + 0.35(100) = 45 + 35 = $80 Both companies charge $80 for a 100-mile rental. Possible answer: For 100 mi, Rapid Rental Car

charges $40 + $15 + $25 = $80. Capital Cars charges $45 + $35 = $80.

b. Capital Cars; the cost would be less.

c. No; now Rapid Rental Car would be cheaper.

d. Less than 100 mi - use Capital Cars; more than 100 mi - use Rapid Rental Car.

53. 3(x + 3) = (x + 3) + (2x - 3) + x 3(x) + 3(3) = (x + 3) + (2x - 3) + x 3x + 9 = x + 3 + 2x - 3 + x 3x + 9 = 4x ________ - 3x ____ - 3x 9 = x

54a. 420 + 60(2) = 420 + 120 = 540 acres

Possible answer: The total acreage in the next 2 days will be about 100 more than 420, so 540 is reasonable.

b. 420 + 60d c. 80d

d. 420 + 60d = 80d _________ - 60d _____ - 60d 420 = 20d

420 ____ 20

= 20d ____ 20

21 days = d d represents the number of days it will take the

firefighters to put out the fire.

55. Possible answer: 2x + 6 = x + 5 + x

56a. 47x 56b. 65x

c. 47x + 1

d. 47x + 1 = 65x - 47x - 47x 1 = 18x

1 ___ 18

= 18x ____ 18

1 ___ 18

= x

The cheetah will have to run for 1 ___ 18

h or about

3 1 __ 3 min to catch the gazelle.

e. No; the cheetah can only maintain its top speed for about 0.003 h, or about 11 s.

57. Possible answer: Simplify both sides if necessary. Collect the variable terms on one side by using inverse operations. Then isolate the variable using inverse operations.

teSt prep

58. C; Lindsey’s monthly magazine costs $1.25 per issue so her total cost is 1.25m. Kenzie got her first two magazines free so she paid for m - 2 magazines. Since Kenzie’s magazines cost $1.50, her total cost is 1.50(m - 2). So when

1.25m = 1.50(m - 2), they will have paid the same amount.

59. F; The equation is 7x = 5x - 3. Solving gives x = -1.5, which is F.

60. B; 3 packs of markers plus $9.00 will be equal to the cost of 5 packs of markers so B is correct.

61. H; In 133 days, or 19 weeks, she will save an additional 20(19) = $380. So she will have 380 + 120 = $500.


CS10_A1_MESK710372_C01.indd 17 3/30/11 10:31:05 PM

62. 4 -2(x - 1) + 5x = 2(2x - 1) -2(x) - 2(-1) + 5x = 2(2x) + 2(-1) -2x +2 + 5x = 4x - 2 3x + 2 = 4x - 2 ________ - 3x ________ - 3x 2 = x - 2 ___ + 2 _____ + 2 4 = x


63. 4x + 2[4 - 2(x + 2)] = 2x - 4 4x + 2(4) + 2[-2(x + 2)] = 2x - 4 4x + 8 - 4(x + 2) = 2x - 4 4x + 8 - 4(x) - 4(2) = 2x - 4 4x + 8 - 4x - 8 = 2x - 4 0 = 2x - 4 ___ + 4 _____ + 4 4 = 2x

4 __ 2 = 2x ___

2

2 = x

64. x + 5 _____ 2 + x - 1 _____

2 = x - 1 _____

3

6 ( x + 5 _____ 2

+ x - 1 _____ 2 ) = 6 ( x - 1 _____

3 )

6 ( x + 5 _____ 2 ) + 6 ( x - 1 _____

2 ) = 6 ( x - 1 _____

3 )

3x + 15 + 3x - 3 = 2x - 2 6x + 12 = 2x - 2 _________ - 2x ________ - 2x 4x + 12 = -2 _______ - 12 ____ - 12 4x = -14

4x ___ 4 = -14 ____

4

x = - 7 __ 2

65. 2 __ 3 w - 1 __

4 = 2 __

3 (w - 1 __

4 )

2 __ 3 w - 1 __

4 = 2 __

3 (w) + 2 __

3 (- 1 __

4 )

2 __ 3 w - 1 __

4 = 2 __

3 w - 1 __

6

_________

- 2 __ 3 w

_________ - 2 __

3 w

- 1 __ 4 = - 1 __

6 7

no solution

66. -5 - 7 - 3f = -f -2(f + 6) -5 - 7 - 3f = -f - 2(f ) - 2(6) -5 - 7 - 3f = -f -2f -12 -12 - 3f = -3f - 12 ________ + 3f ________ + 3f -12 = -12 3 all real numbers

67. 2 __ 3 x + 1 __

2 = 3 __

5 x - 5 __

6

_________

- 3 __ 5 x

________ - 3 __

5 x

1 ___ 15

x + 1 __ 2 = - 5 __

6

________

- 1 __ 2

____ - 1 __

2

1 ___ 15

x = - 4 __ 3

(15) ( 1 ___ 15

x) = (15) (- 4 __ 3 )

x = -20

68. x - 1 __ 4 = x __

3 + 7 3 __

4

_________

- 1 __ 3 x

_________ - 1 __

3 x

2 __ 3 x - 1 __

4 = 7 3 __

4

_______

+ 1 __ 4

____ + 1 __

4

2 __ 3 x = 8

( 3 __ 2 ) ( 2 __

3 x) = ( 3 __

2 ) (8)

x = 12

69. Let n represent the first number. 2(n + 2) = 3n - 2 2(n) + 2(2) = 3n - 2 2n + 4 = 3n - 2 ________ - 2n ________ - 2n 4 = n - 2 ___ +2 _____ + 2 6 = n The numbers are n = 6, n + 1 = 7, and n + 2 = 8.

70. Let n represent the first number. 2n = 12 + n + 2 2n = 14 + n ___ -n ______ - n n = 14 The numbers are n = 14, n + 1 = 15, and

n + 2 = 16.

71. Let r represent the amount of money Rob originally had.

r - 0.80 + 0.38 = r __ 2 - 0.38 + 0.80

r - 0.42 = 1 __ 2 r + 0.42

__________

- 1 __ 2 r

__________ - 1 __

2 r

1 __ 2 r - 0.42 = 0.42

_________ + 0.42 ______ + 0.42

1 __ 2 r = 0.84

(2) ( 1 __ 2 r) = (2)(0.84)

r = 1.68

Rob originally had $1.68.

solVing For A VAriAble

CheCk it OUt!

1. d = rt

d __ r = rt __ r

d __ r = t

t = d __ r

t = 26.2 ____ 18

t ≈ 1.46

It would take Van Dyk 1.46 hours.

2. f = i - gt _____ + gt _____ + gt f + gt = i


1-6

CS10_A1_MESK710372_C01.indd 18 3/30/11 10:31:07 PM

3a. 5 - b = 2t

5 - b _____ 2 = 2t __

2

5 - b _____ 2

= t

b. D = m __ V

(V)(D) = (V) ( m __ V

)

VD = m

VD ___ D

= m __ D

V = m __ D

think and disCUss

1. Possible answer: The formula d = rt is more useful

in the form d __ r = t when you need to determine the

time it takes to travel a certain distance at a certain speed.

2. Possible answer: Isolate w on the right side by subtracting 2ℓ from both sides and then dividing both sides by 2.

3.

P = 4s; s = P _ 4

Common Formulas

Subject

Physical science

Formula

Geometry (perimeter of a

Earth science

square is 4 times the side length)

F = ma; m =

K = C + 273; C = K - 273 (temperature in kelvins is Celsius temperature plus 273)

F _ a

F _ m

; a =

equals mass times acceleration)

(force


1. A literal equation contains more than one variable. A formula shows how to determine the value of one variable when you know the value(s) of one or more other variables. So a formula always contains more than one variable, making it a literal equation.

2a. a = 46c

a ___ 46

= 46c ____ 46

a ___ 46

= c

b. c = a ___ 46

c = 322 ____ 46

c = 7

The room would need 7 circuits.

3. V = ℓwh

V ___ ℓh

= ℓwh ____ ℓh

V ___ ℓh

= w

4. st + 3t = 6 _____ - 3t ____ - 3t st = 6 - 3t

st __ t = 6 - 3t _____

t

s = 6 - 3t _____ t

5. m - 4n = 8 ______ + 4n ____ + 4n m = 8 + 4n

6. f + 4 _____ g = 6

(g) ( f + 4 _____ g ) = (g)(6)

f + 4 = 6g _____ - 4 ___ - 4 f = 6g - 4

7. b + c = 10 ___ a

(a)(b + c) = (a) ( 10 ___ a )

a(b + c) = 10

a(b + c)

_______ (b + c)

= 10 ______

(b + c)

a = 10 _____ b + c


8a. C = 2πr

C ___ 2π

= 2πr ___ 2π

C ___ 2π

= r

b. r = C ___ 2π

r = 15 ___ 2π

r = 7.5 ___ π in.

9. A = P + I ____ - P _______ - P A - P = I

10. -2 = 4r + s ____ - 4r ________ - 4r -2 - 4r = s

11. xy - 5 = k _____ + 5 ___ + 5 xy = k + 5

xy

__ y = k + 5 _____ y

x = k + 5 _____ y

12. m __ n = p - 6

(n) ( m __ n ) = (n)(p - 6)

m = n(p - 6)

m ______ (p - 6)

= n(p - 6)

_______ (p - 6)

m _____ p - 6

= n

13. x - 2 _____ y = z

(y) ( x - 2 _____ y ) = (y)(z)

x - 2 = yz

x - 2 _____ z = yz

__ z

x - 2 _____ z = y

14. S = 180n - 360 _____ + 360 __________ + 360 S + 360 = 180n

S + 360 _______ 180

= 180n _____ 180

S + 360 _______ 180

= n

15. x __ 5 - g = a

_____ + g ___ + g

x __ 5 = a + g

(5) ( x __ 5 ) = (5)(a + g)

x = 5(a + g)

16. A = 1 __ 2

bh

A = h __ 2

b

( 2 __ h ) (A) = ( 2 __

h ) ( h __

2 b)

2A ___ h

= b

17. y = mx + b ___ - b ______ - b y - b = mx

y - b

_____ m = mx ___ m

y - b

_____ m = x

18. a = 3n + 1 ___ - 1 ______ - 1 a - 1 = 3n

a - 1 _____ 3

= 3n ___ 3

a - 1 _____ 3

= n

19. PV = nRT

PV ___ nR

= nRT ____ nR

PV ___ nR

= T

20. T + M = R _____ - M ____ - M T = R - M

21. M = T - R ____ + R _____ + R M + R = T

22. PV = nRT

PV ___ nT

= nRT ____ nT

PV ___ nT

= R


CS10_A1_MESK710372_C01.indd 19 3/30/11 10:31:10 PM

23. 2a + 2b = c _________ - 2a ____ - 2a 2b = c - 2a

2b ___ 2 = c - 2a ______

2

b = c - 2a ______ 2

24. 5p + 9c = p _________ - 5p ____ - 5p 9c = -4p

9c ___ 9 =

-4p ____

9

c = - 4p

___ 9

25. ax + r = 7 ________ - ax ____ - ax r = 7 - ax

26. 3x + 7y = 2 _________ - 3x ____ - 3x 7y = 2 - 3x

7y

___ 7 = 2 - 3x ______

7

y = 2 - 3x ______ 7

27. 4y + 3x = 5 _________ - 4y ____ - 4y 3x = 5 - 4y

3x ___ 3 =

5 - 4y ______

3

x = 5 - 4y

______ 3

28. y = 3x + 3b ____ - 3x _________ - 3x y - 3x = 3b

y - 3x

______ 3 = 3b ___

3

y - 3x

______ 3 = b

29a. Possible answer: t ≈ d ____ 500

b. Possible answer: t ≈ d ____ 500

= 1300 _____ 500

= 2.6

It takes about 2.6 h to fly 1300 mi.

c. Possible answer: t ≈ d ____ 500

(500)(t) ≈ (500) ( d ____ 500

)

500t ≈ d

d. Possible answer: d ≈ 500t = 500(8) = 4000 The airplane can fly about 4000 mi in 8 h.

30. Ei = 9r

Ei __ i = 9r __

i

E = 9r __ i

E = 9(5)

____ 18

E = 2.5

The pitcher’s ERA is 2.5.

31. t = -0.0035a + g ___ - g ____________ - g t - g = -0.0035a

t - g ________

-0.0035 = -0.0035a _________

-0.0035

t - g ________

-0.0035 = a

32. Possible answer: Use inverse operations to isolate the indicated variable on one side of the equation. You must be sure the variable is the only expression on one side of the equation and doesn’t appear on the other side.

33. Possible answer: The variable a appears in 2 terms. Use the Distributive Property to write a - ab as

a(1 - b), and then divide both sides by 1 - b.

34a. Days Acres

1 60

2 120

3 180

4 240

5 300

d 60d

b. A = 60d

c. Extinguishing the Wildfire

0 1 2 3 4 5 6

100

200

300

400

Days A

cres

(5, 300) (4, 240)

(3, 180) (2, 120)

(1, 60)

The graph is a line.

teSt prep

35. C; First, subtract 9 from both sides. Then divide both sides by 3. Simplyfing the fraction gives C.

36. J; To isolate the b variable on the left side, subtract 2a from both sides. Then, since b is multiplied by -5, divide both sides by -5 to undo the multiplication.

37. D; Since Anna knows the length, width, and volume of the box, she need to find the height and therefore needs to solve for h.


38. 3.3x + r = 23.1 _______ - r ____ - r 3.3x = 23.1 - r

3.3x ____ 3.3

= 23.1 - r _______ 3.3

x = 23.1 - r _______ 3.3

39. 2 __ 5 a - 3 __

4 b = c

_______

+ 3 __ 4 b

_____ + 3 __

4 b

2 __ 5 a = c + 3 __

4 b

( 5 __ 2 ) ( 2 __

5 a) = ( 5 __

2 ) (c + 3 __

4 b)

a = 5 __ 2 (c + 3 __

4 b)


CS10_A1_MESK710372_C01.indd 20 3/30/11 10:31:12 PM

40. 3 __ 5 x + 1.4y = 2 __

5

___________

- 3 __ 5 x

_____ - 3 __

5 x

1.4y = 2 __ 5 - 3 __

5 x

1.4y

____ 1.4

= 2 __ 5 - 3 __

5 x _______

1.4

y = 2 __ 5 - 3 __

5 x _______

1.4

41. t = d ____ 500

+ 1 __ 2

____

- 1 __ 2

________ - 1 __

2

t - 1 __ 2 = d ____

500

(500) (t - 1 __ 2 ) = (500) ( d ____

500 )

500t - 250 = d

42. s = 1 __ 2 g t 2

s = t 2 ___

2g

( 2 __ t 2

) (s) = ( 2 __ t 2

) ( t 2 ___

2g )

2s ___ t 2

= g

43. v 2 = u 2 + 2as ____ - u 2 __________ - u 2 v 2 - u 2 = 2as

v 2 - u 2 _______ 2a

= 2as ____ 2a

v 2 - u 2 _______ 2a

= s

44. y = mx + 6 ___ - 6 ______ - 6 y - 6 = mx

y - 6

_____ x = mx ___ x

y - 6

_____ x = m

If m = 0, y - 6 = 0. Therefore y = 6.

45. S = hwft ______ 35,000

( 35,000

______ hwf

) (S) = ( 35,000

______ hwf

) ( hwft ______ 35,000

)

35,000S

_______ hwf

= t

t = 35,000S

_______ hwf

t = 35,000(2370)

___________ 320(144)(15)

t ≈ 120 s

solVing Absolute-VAlue equAtions

CheCk it OUt!

1a. |x| - 3 = 4 _____ + 3 ___ +3 |x| = 7 Case 1 Case 2 x = 7 x = -7

check ___________ |x| - 3 = 4 ____________ |x| - 3 = 4 |7| - 3 4 |-7| - 3 4 7 - 3 4 7 - 3 4 4 4 3 4 4 3

1b. 8 = |x - 2.5| Case 1 8 = x - 2.5

_____ + 2.5 _______ + 2.5 10.5 = x

Case 2

-8 = x - 2.5 _____ + 2.5 _______ + 2.5 -5.5 = x

check ___________ 8 = ⎜x - 2.5⎟ ___________ 8 = ⎜x - 2.5⎟ 8 8 8

⎜10.5 - 2.5⎟ ⎜8⎟ 8 3

8 8 8

⎜-5.5 - 2.5⎟ ⎜-8⎟ 8 3

2a. 2 - |2x - 5| = 7 ____________ -2 ___ -2 -|2x - 5| = 5 (-1)(-|2x - 5|) = (-1)(5) |2x - 5| = -5 7

no solution

2b. -6 + |x - 4| = -6 ___________ +6 ___ +6 |x - 4| = 0 x - 4 = 0 _____ + 4 ___ +4 x = 4

3. First convert millimeters to meters: 180 mm = 0.18 m. Find two numbers that are 0.18 units away from 134 by using the equation ⎜x - 134⎟ = 0.18.Case 1 x - 134 = 0.18 _______ + 134 _____ + 134 x = 134.18

Case 2 x - 134 = -0.18 _______ + 134 _____ + 134 x = 133.82

The minimum height of the bridge is 133.82 m, and the maximum height is 134.18.

think and disCUss

1. Multiply both sides by 5; split into two cases and solve each one.

2.

no solutions:

no solution

one solution: two solutions:

An absolute-value equation can have…

�x + 1� = -1 �x + 1� = 0 -1

�x + 1� = 2 -3, 1


1-7

CS10_A1_MESK710372_C01.indd 21 3/30/11 10:31:14 PM


1. |x| = 6 Case 1 Case 2 x = 6 x = -6

check ________ |x| = 6 _________ |x| = 6 |6| 6 |-6| 6 6 6 3 6 6 3

2. 9 = |x + 5| Case 1 Case 2 9 = x + 5 -9 = x + 5 ___ -5 _____ - 5 ___ -5 _____ - 5 4 = x -14 = x

check _________ 9 = |x + 5| ____________ 9 = |x + 5| 9 |4 + 5| 9 |-14 + 5| 9 |9| 9 |-9| 9 9 3 9 9 3

3. |3x| + 2 = 8 ______ - 2 ___ -2 |3x| = 6 Case 1 Case 2 3x = 6 3x = -6

3x ___ 3 = 6 __

3 3x ___

3 = -6 ___

3

x = 2 x = -2

check ___________ |3x| + 2 = 8 ______________ |3x| + 2 = 8 |3(2)| + 2 8 |3(-2)| + 2 8 |6| + 2 8 |-6| + 2 8 6 + 2 8 6 + 2 8 8 8 3 8 8 3

4. 2|x| = 18

2|x| ___ 2 = 18 ___

2

|x| = 9 Case 1 Case 2 x = 9 x = -9

check _________ 2|x| = 18 ____________ 2|x| = 18 2|9| 18 2|-9| 18 2(9) 18 2(9) 18 18 18 3 18 18 3

5. ⎜x + 1 __ 2 ⎟ = 1

Case 1

x + 1 __ 2 = 1

_____

- 1 __ 2

____ - 1 __

2

x = 1 __ 2

Case 2

x + 1 __ 2 = -1

_____

- 1 __ 2

____ - 1 __

2

x = - 3 __ 2

check

__________

⎜x + 1 __ 2 ⎟ = 1

____________ ⎜x + 1 __

2 ⎟ = 1

⎜ 1 __ 2 + 1 __

2 ⎟

⎜1⎟ 1

1

1 1 3

⎜ 3 __ 2 + 1 __

2 ⎟

⎜1⎟ 1

1

1 1 3

6. ⎜x - 3⎟ - 6 = 2 __________ + 6 ___ + 6 ⎜x - 3⎟ = 8

Case 1 x - 3 = 8 _____ + 3 ___ + 3 x = 11

Case 2 x - 3 = -8 _____ + 3 ___ + 3 x = -5

check ______________ ⎜x - 3⎟ - 6 = 2 ______________ ⎜x - 3⎟ - 6 = 2

⎜11 - 3⎟ - 6 ⎜8⎟ - 6 8 - 6 2

2 2 2 2 3

⎜-5 - 3⎟ - 6 ⎜8⎟ - 6 8 - 6 2

2 2 2 2 3

7. -8 = |x| 7 no solution

8. |x| = 0 x = 0

check ________ |x| = 0

|0| 0 0 0 3

9. |x + 4| = -7 7 no solution

10. 7 = |3x + 9| + 7

___ -7 __________ - 7 0 = |3x + 9| 0 = 3x + 9 ___ -9 ______ - 9 -9 = 3x

-9 ___ 3 = 3x ___

3

-3 = x

check _________________ 7 = |3x + 9| + 7 7 |3(-3) + 9| + 7 7 |-9 + 9| + 7 7 |0| + 7 7 0 + 7 7 7 3


CS10_A1_MESK710372_C01.indd 22 3/30/11 10:31:15 PM

11. ⎜2.8 - x⎟ + 1.5 = 1.5 - 1.5 - 1.5 ⎜2.8 - x⎟ = 0 2.8 - x = 0 _____________ + x ___ + x 2.8 = xcheck

⎜2.8 - x⎟ + 1.5 = 1.5

⎜2.8 - 2.8⎟ + 1.5 0 + 1.5 1.5

1.5 1.5 1.5 3

12. 5|x + 7| + 14 = 8 ___________ - 14 ____ -14 5|x + 7| = -6

5|x + 7| _______ 5 = -6 ___

5

|x + 7| = - 6 __ 5 7

no solution

13. Let x represent the mile-marker number. |x - 207| = 2 Case 1 Case 2 x - 207 = 2 x - 207 = -2 ______ + 207 _____ +207 _______ + 207 _____ +207 x = 209 x = 205

The walkie-takie will reach between mile marker 205 and mile marker 209.


14. ⎜x⎟ = 1 __ 5

Case 1

x = 1 __ 5

Case 2

x = - 1 __ 5

check

_______

x = 1 __ 5

________ x = - 1 __

5

⎜ 1 __ 5 ⎟

1 __ 5

1 __ 5

1 __ 5 3

⎜ 1 __ 5 ⎟

1 __ 5

1 __ 5

1 __ 5 3

15. |2x - 4| = 22 Case 1 Case 2 2x - 4 = 22 2x - 4 = -22 _____ + 4 ___ +4 ______ + 4 ____ +4 2x = 26 2x = -18

2x ___ 2 = 26 ___

2 2x ___

2 = -18 ____

2

x = 13 x = -9

check _______________ |2x - 4| = 22 _______________ |2x - 4| = 22 |2(13) - 4| 22 |2(-9) - 4| 22 |26 - 4| 22 |-18 - 4| 22 |22| 22 |-22| 22 22 22 3 22 22 3

16. 18 = 3|x - 1|

18 ___ 3 = 3|x - 1| _______

3

6 = |x - 1| Case 1 Case 2 6 = x - 1 -6 = x - 1 ___ +1 _____ + 1 ___ +1 _____ + 1 7 = x -5 = x

check ____________ 18 = 3|x - 1| _____________ 18 = 3|x - 1| 18 3|7 - 1| 18 3|-5 - 1| 18 3|6| 18 3|-6| 18 3(6) 18 3(6) 18 18 3 18 18 3

17. -2|x| = -4

-2|x| _____ -2

= -4 ___ -2

|x| = 2 Case 1 Case 2 x = 2 x = -2

check ___________ -2|x| = -4 _____________ -2|x| = -4 -2|2| -4 -2|-2| -4 -2(2) -4 -2(2) -4 -4 -4 3 -4 -4 3

18. 3|x| - 12 = 18 _______ + 12 ____ +12 3|x| = 30

3|x| ___ 3 = 30 ___

3

|x| = 10 Case 1 Case 2 x = 10 x = -10

check _______________ 3|x| - 12 = 18 _________________ 3|x| - 12 = 18 3|10| - 12 18 3|-10| - 12 18 3(10) - 12 18 3(10) - 12 18 30 - 12 18 30 - 12 18 18 18 3 18 18 3

19. |x - 42.04| = 23.24 Case 1 Case 2 x - 42.04 = 23.24 x - 42.04 = -23.24 ________ + 42.04 ______ +42.04 ________ + 42.04 ______ +42.04 x = 65.28 x = 18.8

check _____________________ |x - 42.04| = 23.24 _____________________ |x - 42.04| = 23.24 |65.28 - 42.04| 23.24 18.8 - 42.04| 23.24 |23.24| 23.24 |-23.24| 23.24 23.24 23.24 3 23.24 23.24 3


CS10_A1_MESK710372_C01.indd 23 3/30/11 10:31:16 PM

20. ⎜ 2 __ 3 x - 2 __

3 ⎟ = 2 __

3

Case 1 Case 2

2 __ 3 x - 2 __

3 = 2 __

3 2 __

3 x - 2 __

3 = - 2 __

3

______

+ 2 __ 3

___ + 2 __

3

_______ + 2 __

3

___ + 2 __

3

2 __ 3 x = 4 __

3 2 __

3 x = 0

( 3 __ 2 ) 2 __

3 x = ( 3 __

2 ) 4 __

3 ( 3 __

2 ) 2 __

3 x = ( 3 __

2 ) 0

x = 2 x = 0

check

______________

⎜ 2 __ 3 x - 2 __

3 ⎟ = 2 __

3

_______________ ⎜ 2 __

3 x - 2 __

3 ⎟ = 2 __

3

⎜ 2 __ 3 (2) - 2 __

3 ⎟ 2 __

3 ⎜ 2 __

3 (0) - 2 __

3 ⎟ 2 __

3

⎜ 4 __ 3 - 2 __

3 ⎟ 2 __

3 ⎜0 - 2 __

3 ⎟ 2 __

3

⎜ 2 __ 3 ⎟ 2 __

3 ⎜- 2 __

3 ⎟ 2 __

3

2 __ 3 2 __

3 3 2 __

3 2 __

3 3

21. |3x + 1| = 13 Case 1 Case 2 3x + 1 = 13 3x + 1 = -13 _____ - 1 ___ -1 ______ - 1 ____ -1 3x = 12 3x = -14

3x ___ 3 = 12 ___

3 3x ___

3 = -14 ____

3

x = 4 x = - 14 ___ 3

check _____________ |3x + 1| = 13 |3(4) + 1| 13 |12 + 1| 13 |13| 13 13 13 3

_________________ |3x + 1| = 13

⎜3 (- 14 ___ 3 ) + 1⎟ 13

|-14 + 1| 13 |-13| 13 13 13 3

22. ⎜-2x + 3⎟ = 5.8Case 1 -2x + 3 = 5.8 _______ - 3 ___ - 3 -2x = 2.8

-2x ____ -2

= 2.8 ___ -2

x = -1.4

Case 2 -2x + 3 = -5.8 _______ - 3 ___ - 3 -2x = -8.8

-2x ____ -2

= -8.8 _____ -2

x = 4.4check

⎜-2x + 3⎟ = 5.8 ⎜-2x + 3⎟ = 5.8

⎜-2(-1.4) + 3⎟ ⎜2.8 + 3⎟ ⎜5.8⎟ 5.8

5.8 5.8 5.8 5.8 3

⎜-2(4.4) + 3⎟ ⎜-8.8 + 3⎟ ⎜-5.8⎟ 5.8

5.8 5.8 5.8 5.8 3

23. ⎜4x⎟ + 9 = 9 - 9 - 9 ⎜4x⎟ = 0 4x = 0

4x ___ 4 = 0 __

4

x = 0check ____________ ⎜4x⎟ + 9 = 9

⎜4(0)⎟ + 9 ⎜0⎟ + 9 0 + 9 9

9 9 9 9 3

24. 8 = 7 - |x| ___ -7 _______ -7 1 = -|x| (-1)(1) = (-1)(-|x|) - 1 = |x| 7 no solution

25. |x| + 6 = 12 - 6 |x| + 6 = 6 _____ - 6 ___ -6 |x| = 0 x = 0

check ______________ |x| + 6 = 12 - 6 |0| + 6 6 0 + 6 6 6 6 3

26. ⎜x - 3⎟ + 14 = 5 ___________ - 14 ____ - 14 ⎜x - 3⎟ = -9 7no solution

27. 0 = ⎜ 2 __ 3 - x⎟

0 = 2 __ 3 - x

___ + x ______ + x

x = 2 __ 3

check

__________

0 = ⎜ 2 __ 3 - x⎟

0

0 0

⎜ 2 __ 3 - 2 __

3 ⎟

⎜0⎟ 0 3

28. 3 + ⎜x - 1⎟ = 3 ____________ - 3 ___ - 3 ⎜x - 1⎟ = 0 x - 1 = 0 ____________ + 1 ___ + 1 x = 1check _____________ 3 + ⎜x - 1⎟ = 3 3 + ⎜1 - 1⎟ 3 + ⎜0⎟ 3 + 0 3

3 3 3 3 3

29. Let x represent the diameter of the valve. |x - 5| = 0.001 Case 1 Case 2 x - 5 = 0.001 x - 5 = -0.001 ____ + 5 ______ +5 _____ + 5 ______ +5 x = 5.001 x = 4.999

The value can be between 4.999 mm and 5.001 mm.


CS10_A1_MESK710372_C01.indd 24 3/30/11 10:31:18 PM

30. |n - 3| = 5 Case 1 Case 2 n - 3 = 5 n - 3 = -5 _____ + 3 ___ +3 _____ + 3 ___ +3 n = 8 n = -2

6 4 2 8 0 -2 -4

31. |x - 7| = 2 Case 1 Case 2 x - 7 = 2 x - 7 = -2 ____ + 7 ___ +7 _____ + 7 ___ +7 x = 9 x = 5

5 6 8 7 9

32. Let x represent the diameter. ⎜x - 6.5⎟ = 0.04Case 1 x - 6.5 = 0.04 ______ + 6.5 _____ + 6.5 x = 6.54

Case 2 x - 6.5 = -0.04 ______ + 6.5 _____ + 6.5 x = 6.46

Maximum diameter is 6.54 mm; minimum diameter is 6.46 mm.

33. Let x represent the number of bricks. |x - 1500| = 0.05(1500) |x - 1500| = 75 Case 1 Case 2 x - 1500 = 75 x - 1500 = -75 _______ + 1500 ______ +1500 ________ + 1500 ______ +1500 x = 1575 x = 1425

The maximum number of bricks is 1575, and the minimum number of bricks is 1425.

34. ⎜ℓ - 65.1⎟ = 0.2Case 1 ℓ - 65.1 = 0.2 _______ + 65.1 ______ + 65.1 ℓ = 65.3

Case 2 ℓ - 65.1 = -0.2 _______ + 65.1 ______ + 65.1 ℓ = 64.9

No; acceptable range is from 64.9 in. to 65.3 in.

35. 3 - (-3)

________ 2 = 6 __

2 = 3

3 - 3 = 0 Equation is ⎜x - 0⎟ = 3 ⎜x⎟ = 3

36. 1 - (-1)

________ 2 = 2 __

2 = 1

1 - 1 = 0 Equation is ⎜x - 0⎟ = 1 ⎜x⎟ = 1

37. 5 - (-1)

________ 2 = 6 __

2 = 3

5 - 3 = 2 Equation is ⎜x - 2⎟ = 3

38. 2 - (-4)

________ 2 = 6 __

2 = 3

2 - 3 = -1 Equation is ⎜x - (-1)⎟ = 3 ⎜x + 1⎟ = 3

39. sometimes; if equation simplifies to an absolute value on one side and a positive number on the other

40. sometimes; if x is non-negative

41. always; if x is non-negative, ⎜x⎟ = x is non-negative, and if x is negative, ⎜x⎟ = -x, which is non-negative.

42. Let x represent the temperature of the freezer. |x - 2| = 2.5 Case 1 Case 2 x - 2 = 2.5 x - 2 = -2.5 ____ + 2 ____ +2 _____ + 2 _____ +2 x = 4.5 x = -0.5

The maximum temperature in the freezer is 4.5˚F, and the minimum temperature in the freezer is -0.5˚F.

43a. |t - 24| = 5

b. Case 1 Case 2 t - 24 = 5 t - 24 = -5 _____ + 24 ____ +24 _____ + 24 ____ +24 t = 29 t = 19

c. yes, if you change the left side so that the measured wind speed is being subtracted from t

d. The measurement is correct within 5 mi/h.

44a. ⎜r - 55⎟

b. ⎜r - 55⎟ = 45

c. Case 1 r - 55 = 45 ______ + 55 ____ +55 r = 100

Case 2 r - 55 = -45 ______ + 55 ____ + 55 r = 10

Least: 10 gal/min Greatest: 100 gal/min

45. Possible answer: No; when an absolute-value expression equals 0, there is only one equation to solve. When an absolute-value expression equals a negative number, there is no equation to solve.

46. No; no matter what value of a is chosen, there will always be two solutions: a + 1 and a - 1.

teSt prep

47. C; Maximum and mimimum weights are (65 + 3) kg and (65 - 3) kg, respectively.

48. J; one case is n + 1 = -2 _____ - 1 ___ - 1 n = -3

49. B; because 95 - 90 _______ 2 = 2.5 and 95 - 2.5 = 92.5


CS10_A1_MESK710372_C01.indd 25 3/30/11 10:31:20 PM


50. P = 2ℓ + 2w 100 = 2 ⎜2x - 4⎟ + 2x ________ - 2x _____________ - 2x 100 - 2x = 2 ⎜2x - 4⎟ Case 1 100 - 2x = 2(2x - 4) 100 - 2x = 2(2x) - 2(4) 100 - 2x = 4x - 8 _________ + 8 ______ + 8 108 - 2x = 4x ________ + 2x ____ + 2x 108 = 6x

108 ____ 6 = 6x ___

6

18 = x Case 2 -(100 - 2x) = 2(2x - 4) -100 - (-2x) = 2(2x) - 2(4) -100 + 2x = 4x - 8 ____________ + 8 _______ + 8 -92 + 2x = 4x ____________ - 2x ____ - 2x -92 = 2x

-92 ____ 2 = 2x ___

2

-46 = x The value of x is 18 in.; solutions to equation are 18 and -46, but since x represents length, the negative solution is not reasonable.

51. Div. Prop. of Eq.; Subtr. Prop. of Eq.; Div. Prop. of Eq.

52. ⎜x⎟ = ⎜x + 1⎟ Case 1 x ≥ 0 x = x + 1 ___ - x ______ - x 0 = 1 7 no solution

Case 2 x < 0 if x + 1 < 0, then -x = -(x + 1) no solution as in Case 1 if x + 1 ≥ 0, then -x = x + 1 ____ - x ______ - x -2x = 1

-2x ____ -2

= 1 ___ -2

x = - 1 __ 2

check ⎜x⎟ = ⎜x + 1⎟

⎜- 1 __ 2 ⎟

1 __ 2

1 ____ 2

⎜- 1 __ 2 + 1⎟

⎜ 1 __ 2 ⎟

1 __ 2 3

reAdY to go on? section A quiz

1 – 4. Possible answers given:

1. the sum of 4 and n; n increased by 4

2. the difference of m and 9; 9 less than m

3. g divided by 2; the quotient of g and 2

4. 4 times z; the product of 4 and z

5. x - 32 = -18 _____ + 3 _ 2 ____ + 32 x = 14

6. 1.1 = m - 0.9 _____ + 0.9 _______ + 0.9 2 = m

7. j + 4 = -17 ____ ____ - 4 ____ - 4 j = -21

8. 9 __ 8 = g + 1 __

2

____

- 1 __ 2

______ - 1 __

2

5 __ 8 = g

9. Let s represent the space used on Soledad’s hard drive.

313 + s = 400 _________ - 313 _____ - 313 s = 87 Soledad used 87 GB of hard drive space in the first

six months.

10. h __ 3 = -12

(3) ( h __ 3 ) = (3)(-12)

h = -36

11. -2.8 = w ___ -3

(-3)(-2.8) = (-3) ( w ___ -3

)

8.4 = w

12. 42 = 3c

42 ___ 3 = 3c ___

3

14 = c

13. -0.1b = 3.7

-0.1b ______ -0.1

= 3.7 _____ -0.1

b = -37

14. Let g represent the amount of the goal.

3 __ 5 g = 2400

( 5 __ 3 ) ( 3 __

5 g) = ( 5 __

3 ) (2400)

g = 4000

The fund-raising goal was $4000.

15. 2r + 20 = 200 ______ - 20 ____ - 20 2r = 180

2r __ 2 = 180 ____

2

r = 90

16. 3 __ 5 k + 5 = 7

______ - 5 ___ - 5

3 __ 5 k = 2

( 5 __ 3 ) ( 3 __

5 k) = ( 5 __

3 ) (2)

k = 10 ___ 3

17. 5n + 6 - 3n = -12 2n + 6 = -12 ______ - 6 ____ - 6 2n = -18

2n ___ 2 = -18 ____

2

n = -9

18. 4(x - 7) = 2 4(x) + 4(-7) = 2 4x - 28 = 2 _______ + 28 ____ + 28 4x = 30

4x ___ 4 = 30 ___

4

x = 7.5


CS10_A1_MESK710372_C01.indd 26 3/30/11 10:31:21 PM

19. Let m represent the number of miles Carmen traveled.

2.10 + 0.80m = 11.70 _____________ - 2.10 ______ - 2.10 0.80m = 9.6

0.80m ______ 0.80

= 9.6 ____ 0.80

m = 12 Carmen traveled 12 miles.

20. 4x - 3 = 2x + 5 ________ - 2x ________ - 2x 2x - 3 = 5 ______ + 3 ___ + 3 2x = 8

2x ___ 2 = 8 __

2

x = 4

21. 3(2x - 5) = 2(3x - 2) 3(2x) + 3(-5) = 2(3x) + 2(-2) 6x - 15 = 6x - 4 _________ - 6x ________ - 6x -15 = -4 7 no solution

22. 2(2t - 3) = 6(t + 2) 2(2t) + 2(-3) = 6(t) + 6(2) 4t - 6 = 6t + 12 _______ - 4t ________ - 4t -6 = 2t + 12 ____ - 12 ______ - 12 -18 = 2t

-18 ____ 2 = 2t __

2

-9 = t

23. 7(x + 5) = -7(x + 5) 7(x) + 7(5) = -7(x) - 7(5) 7x + 35 = -7x - 35 _________ + 7x _________ + 7x 14x + 35 = -35 _______ - 35 ____ - 35 14x = -70

14x ____ 14

= -70 ____ 14

x = -5

24. 2x + 3y = 12 ______ - 3y ____ - 3y 2x = 12 - 3y

2x ___ 2 =

12 - 3y _______

2

x = 12 - 3y

_______ 2

x = 6 - 3 __ 2 y

25. x __ r = v

(r) ( x __ r ) = (r)(v)

x = rv

26. 5j + s = t - 2 _____ + 2 ____ + 2 5j + s + 2 = t

27. h + p = 3(k - 8)

h + p

_____ 3

=

3(k - 8) _______

3

h + p

_____ 3

= k - 8

_____ + 8 _____ + 8

h + p

_____ 3

+ 8 = k

28. ⎜r⎟ = 7

Case 1 Case 2

r = -7 r = 7 29. ⎜h + 4⎟ = 11

Case 1 Case 2

h + 4 = -11 h + 4 = -11

_____ -4 ___ -4 _____ -4 ___ -4

h = -15 h = 7

30. ⎜2x + 4⎟ = 0

2x + 4 = 0

______ -4 ___ -4

2x = -4

2x ___ 2 = -4 ___

2

x = -2

31. 16 = 7 ⎜p + 3⎟ + 30

____ -30 __________ -30

-14 = 7 ⎜p + 3⎟

-14 ____ 7 =

7 ⎜p + 3⎟ ________

7

-2 = ⎜p + 3⎟ no solution

rAtes, rAtios, And proportions

CheCk it OUt!

1. 18 ÷ 3 __ 2

= 18 × 2 __ 3

= 12

2. 52.50 ÷ 7 = $7.50/h

3. 56 mi _____ 4 h

· 5280 ft ______ mi

· h _______ 3600 s

= 20.5 ̶

3 ft/s ≈ 20.5 ft/s

Possible answe r: The cyclist travels about 60 mi in 4 h, or 15 mi/h. There are about 5000 ft in 1 mi, so this speed is equivalent to 15(5000) = 75,000 ft/h. There are about 4000 s in 1 h, so this is equivalent

to 75,000

______ 4000

≈ 20 ft/s. So 20.5 ft/s is reasonable.


1-8

CS10_A1_MESK710372_C01.indd 27 3/30/11 10:31:23 PM

4a. -5 ___ 2 =

y __

8

y = - 5 __ 2 × 8

y = -20

4b. g + 3

______ 5

= 7 __ 4

g + 3 = 7 __ 4 × 5

g = 35 ___ 4 - 12 ___

4

g = 23 ___ 4 = 5.75

5. 16 ___ x = 32 ___ 1

x = 16 ___ 32

x = 0.5 ft = 6 in.

think and disCUss

1. Possible answer: by using cross products; by multiplying both sides by 4 to isolate t.

2. Possible answer: 1 in. on the map represents 18 mi. So 0.625 in. will represent a little more than half of 18 mi. So 11.25 is reasonable.

3. Possible answers: rate: to compare quantities with different units; unit rate: to compare prices of products of different sizes; proportion: to solve for a missing quantity; conversion factor: to convert from one set of units to another.

Uses of Ratios

Proportion: to solve for a missing quantity

Unit rate: to compare prices of products of different sizes

Conversion factor: to convert from one set of units to another

Scale: to make scale drawings/models or to read distances from a map

Rate: to compare quantities with different units


1. The ratios are equivalent.

2. x ____ $64

= 3 __ 4

x = 3 __ 4 × $64

x = $48

3. x _________ 341 trillion

= 2 __ 1

x = 2 __ 1 × 341 trillion

= 682 trillion

4. x __ 1 = 2000 _____

40

x = 50 rotations/s

5. x __ 1 =

224,988 _______

12

x = 18,749 lb/cow

6. x __ 1 = 4.5 ___

9

x = 0.5 m/s

7. 4 1 __ 2 /h · 1 h ______

60 min

= 0.075 page/min

8. 18 ft ____ 2 s

· 3600 s ______ 1 h

· 1 mi ______ 5280 ft

≈ 6.14 mi/h

9. 125 yards

___________ 1 tablespoon

· 256 tablespoons

______________ 1 gallon

· 1 mi __________ 1760 yards

= 18 mi/gal

10. 3 __ z = 1 __ 8

1(z) = 3(8) z = 24

11. x __ 3 = 1 __

5

5(x) = 1(3) 5x = 3

x = 3 __ 5

12. b __ 4 = 3 __

2

2(b) = 4(3) 2b = 12 b = 6

13. f + 3 _____ 12

= 7 __ 2

2(f + 3) = 12(7) 2f + 6 = 84 ____ - 6 ___ - 6 2f = 78 f = 39

14. -1 ___ 5 = 3 ___

2d

-1(2d) = 5(3) -2d = 15 d = -7.5

15. 3 ___ 14

= s - 2 _____ 21

14(s - 2) = 21(3)14s - 28 = 63 _______ + 28 ____ + 28 14s = 91 s = 6.5

16. -4 ___ 9 = 7 __ x

-4(x) = 7(9) -4x = 63 x = -15.75

17. 3 _____ s - 2

= 1 __ 7

1(s - 2) = 3(7) s - 2 = 21 _____ + 2 ____ + 2 s = 23

18. 10 ___ h = 52 ___

13

52(h) = 10(13) 52h = 130 h = 2.5

19. Let h be the height of the Altar Stone.

3 __ 5 = h ___

4.9

5(h) = 4.9(3) 5h = 14.7 h = 2.94 m


20. 6 __ x = 1 __ 9

1(x) = 6(9) x = 54 in.

21. x ______ 12,000

= 3 ____ 500

500(x) = 12,000(3) 500x = 36,000 x = 72

22. 25 lb _____ 4 gal

= 6.25 lb/gal

23. $6058.50

________ 15 oz

= $403.90/oz

24. 11.9 ft ______ 3 days

= 11.9 ft ______ 3 days

· 1 day

_____ 24 h

· 12 in. _____ 1 ft

≈ 1.98 in./h11.9 ft in 3 days is roughly equivalent to 12 ft

in 72 h, or 12 ___ 72

= 1 __ 6 ft/h. There are 12 in.

in 1 ft, so this speed is equivalent to 12 ___ 6 = 2 in./h.

So 1.98 in./h is reasonable.


CS10_A1_MESK710372_C01.indd 28 3/30/11 10:31:24 PM

25. 694 m/s

= 694 m ______ 1 s

· 3600 s ______ 1 h

· 1 km _______ 1000 m

= 2498.4 km/h

26. v __ 6 = 1 __

2

2(v) = 6(1)

2v ___ 2 = 6 __

2

v = 3

27. 2 __ 5 = 4 __ y

2(y) = 4(5)

2y

___ 2 = 20 ___

2

y = 10

28. 2 __ h = -5 ___

6

(-5)(h) = 2(6)

(-5)h

_____ (-5)

= 12 ____

(-5)

h = -2.4

29. 3 ___ 10

= b + 7 _____ 20

10(b + 7) = 20(3) 10b + 70 = 60 _______ - 70 ____ - 70 10b = -10

10b ____ 10

= -10 ____ 10

b = -1

30. 5t __ 9 = 1 __

2

2(5t) = 1(9)

10t ___ 10

= 9 ___ 10

t = 0.9

31. 2 __ 3 = 6 _____

q - 4

2(q - 4) = 3(6) 2q - 8 = 18 _____ + 8 ____ + 8 2q = 26

2q

___ 2 = 26 ___

2

q = 13

32. x __ 8 = 7.5 ___

20

20(x) = 8(7.5)

20x ____ 20

= 60 ___ 20

x = 3

33. 3 __ k = 45 ___

18

45(k) = 3(18)

45k ____ 45

= 54 ___ 45

k = 1.2

34. 6 __ a = 15 ___ 17

15(a) = 6(17)

15a ____ 15

= 102 ____ 15

a = 6.8

35. 9 __ 2 = 5 _____

x + 1

9(x + 1) = 2(5) 9x + 9 = 10 ______ - 9 ___ - 9 9x = 1

9x ___ 9 = 1 __

9

x = 1 __ 9

36. 3 __ 5 = x ____

100

5(x) = 3(100)

5x ___ 5 = 300 ____

5

x = 60

37. 38 ___ 19

= n - 5 _____ 20

19(n - 5) = 20(38) 19n - 95 = 760 ______ + 95 ____ + 95 19n = 855

19n ____ 19

= 855 ____ 19

n = 45

38. 100 ____ 1 = 30 mm ______ x

100x = 30 mm x = 0.3 mmSo, the dust mite is 0.3 mm long.

39. x ___ 70

= 60 ___ 50

50(x) = 60(70)

50x ____ 50

= 4200 _____ 50

x = $8470 euro will be worth more than $60, so $84 is reasonable.

40. 20 ___ 85

= 100 ____ c

20(c) = 100(85)

20c ____ 20

= 8500 _____ 20

c = 425 carp

41. Possible answer: The denominator in the first ratio is the number of varsity members, but in the second ratio it is the total number of people on the team.

42a. v = d __ t

100 m ______ 10.5 s

≈ 9.52 m/s 200 m ______ 21.3 s

≈ 9.39 m/s

800 m ______ 113.3 s

≈ 7.06 m/s 5000 m _______ 864.7 s

≈ 5.78 m/s

b. The 100 m race has the fastest speed, whereas, the 5000 m race has the slowest speed.

c. Possible answer: Runners can maintain a very high speed for a short distance, but over a longer distance their speed drops.

43. 216 frames - 203 frames _____________________ 8 hours

= 1.625 frames/h

44. x - 1 _____ 3

= x + 1 _____ 5

5(x - 1) = 3(x + 1) 5x - 5 = 3x + 3 ________ - 3x + 5 ________ - 3x + 5 2x = 8

2x ___ 2 = 8 __

2

x = 4

45. m __ 3

= m + 4 ______ 7

7(m) = 3(m + 4) 7m = 3m + 12 _____ - 3m _________ - 3m 4m = 12

4m ___ 4

= 12 ___ 4

m = 3

46. 1 _____ x - 3

= 3 _____ x - 5

1(x - 5) = 3(x - 3) x - 5 = 3x - 9 ________ - 3x + 5 ________ - 3x + 5 -2x = -4

-2x ____ -2

= -4 ___ -2

x = 2

47. a __ 2

= a - 4 _____ 30

30(a) = 2(a - 4) 30a = 2a - 8 ____ - 2a ________ - 2a 28a = -8

28a ____ 28

= -8 ___ 28

a = - 2 __ 7


CS10_A1_MESK710372_C01.indd 29 3/30/11 10:31:26 PM

48. 3 ___ 2y

= 16 _____ y + 2

3(y + 2) = 16(2y) 3y + 6 = 32y ________ - 3y ____ - 3y 6 = 29y

6 ___ 29

= 29y

____ 29

y = 6 ___ 29

49. n + 3 _____ 5

= n - 1 _____ 2

2(n+3) = 5(n - 1) 2n + 6 = 5n - 5 ________ - 5n - 6 ________ - 5n - 6 -3n = -11

-3n ____ -3

= -11 ____ -3

n = 11 ___ 3

50. 1 __ y = 1 ______ 6y - 1

1(6y - 1) = 1(y) 6y - 1 = y _______ - y + 1 _______ - y + 1 5y = 1

5y

___ 5

= 1 __ 5

y = 1 __ 5

51. 2 __ n = 4 _____ n + 3

2(n + 3) = 4(n) 2n + 6 = 4n ________ - 4n - 6 ________ - 4n - 6 -2n = -6

-2n ____ -2

= -6 ___ -2

n = 3

52. 5t - 3 ______ -2

= t + 3 _____ 2

2(5t - 3) = -2(t + 3) 10t - 6 = -2t - 6 _______ + 2t + 6 _______ + 2t + 6 12t = 0 t = 0

53. 3 _____ d + 3

= 4 ______ d + 12

3(d + 12) = 4(d + 3) 3d + 36 = 4d + 12 _________ - 4d - 36 _________ - 4d - 36 -d = -24

-d ___ -1

= -24 ____ -1

d = 24

54. 3x + 5 ______ 14

= x __ 3

3(3x + 5) = 14(x) 9x + 15 = 14x __________ - 14x - 15 __________ - 14x - 15 -5x = -15

-5x ____ -5

= -15 ____ -5

x = 3

55. 5 ___ 2n

= 8 _______ 3n - 24

5(3n - 24) = 8(2n) 15n - 120 = 16n ___________ - 16n + 120 ___________ - 16n + 120 -n = 120

-n ___ -1

= 120 ____ -1

n = -120

ApplicAtions oF proportions

CheCk it OUt!

1. 7 __ 5 = x __

2

5(x) = 7(2)

5x ___ 5 = 14 ___

5

x = 2.8 in.

2a. 150 ____ x = 45 ____ 195

45(x) = 150(195)

45x ____ 45

= 29,250

______ 45

x = 650 cm

b. 5.5 ___ x = 3.5 ___ 28

3.5(x) = 5.5(28)

3.5x ____ 3.5

= 154 ____ 3.5

x = 44 ft

3. The ratio of the perimeters is equal to the ratio of the corresponding sides.

think and disCUss

1. Possible answer: a model airplane and the real airplane it represents; a baseball and a softball; a photograph and its enlargement

2. �ABC ∼ �DEF

AB corresponds to DE. BC corresponds to EF. AC corresponds to DF.

Angle A corresponds to angle D. Angle B corresponds to angle E. Angle C corresponds to angle F.

Corresponding angles:

A

B E

D

FC

Corresponding sides:


1. They are the same shape but not necessarily the same size.

2. 5 ___ 4 = 7 __ x

5x = 28

5x ___ 5 = 28 ___

5

x = 5.6 m

3. 4 __ 8 = 5 __ x

4x = 40

4x ___ 4 = 40 ___

4

x = 10 ft

4. 5 __ h = 3.5 ___

14

3.5h = 70

3.5h ____ 3.5

= 70 ___ 3.5

h = 20 ft

5. The ratio of the areas is the square of the ratio of the corresponding side lengths.


6. 2 __ 4 = 4 __ x

2x = 16

2x ___ 2 = 16 ___

2

x = 8 m

7. 2 ___ 10

= x ___ 35

10x = 70

10x ____ 10

= 70 ___ 10

x = 7 in.

8. 18 ____ h = 20 ___

42

20h = 756

20h ____ 20

= 756 ____ 20

h = 37.8 ft

9. The ratio of the perimeters is equal to the ratio of the corresponding side lengths.


1-9

CS10_A1_MESK710372_C01.indd 30 3/30/11 10:31:28 PM

10. 2 1 __

2 ___

2 =

6 1 __ 4 ___ x

2 1 __ 2 x = 12 2 __

4

2 1 __

2 x ____

2 1 __ 2 =

12 2 __ 4 ____

2 1 __ 2

x = 5 ft

Possible answer: The width is less than the length for the baby blanket, so the width on the mother’s blanket will also be less than the length. Since 5 ft is

less than 6 1 __ 4 ft, 5 ft is a reasonable answer.

11. 120 ____ x = ( 1 __ 2 )

2

120 ____ x = 1 __ 4

x = 480 ft 2

12. 64 ___ 16

= ( x __ 1 )

2

64 ___ 16

= x 2 __ 1

x 2 = 4 x = 2

13. 6272 _____ 98

= ( x __ 1 )

3

64 = x 3 x = 4

14. 12 ___ x = 8 __ 6

8x = 72

8x ___ 8 = 72 ___

8

x = 9 m

15. 14 ___ x = 10 ___ 2

10x = 28

10x ____ 10

= 28 ___ 10

x = 2.8 ft

16. 7 ___ x = 8 ___ 10

8x = 70

8x ___ 8 = 70 ___

8

x = 8.75 ft

17. 8 __ x = 3 + 3 _____ 3

6x = 24

6x ___ 6 = 24 ___

6

x = 4 cm

18. 4 ___ x = 6 ____ 450

6x = 1800

6x ___ 6 = 1800 _____

6

x = 300 ft

19. Possible answer: No; a 16-in. pizza actually has 4 times the area, so the cost should be 4 times as much.

20a. 12 ____ 100

; 18 ____ 100

; 25 ____ 100

; 67 ____ 100

; 98 ____ 100

b. 0.12; 0.18; 0.25; 0.67; 0.98

c. The decimal is the percent with its decimal point moved two places to the left and the % symbol removed.

21. 1.5 ___ x = 4.5 ___ 36

4.5x = 54

4.5x ____ 4.5

= 54 ___ 4.5

x = 12 m

22. 40 ____ x = 30 ___ 72

30x = 2880

30x ____ 30

= 2880 _____ 30

x = 96 m

23. k 2

teSt prep

24. C; The radius of the first ball is twice the radius of the second. Since 2 3 = 8, to find the volume of the smaller sphere, divide 800 by 8.

25. G; Students can look at the first ratio in each similarity statement and check that the letters S and G are in the same position as M and W. Only choice G has them placed incorrectly.

26. 3 __ x = 5 ____ 7.25

5x = 21.75

5x ___ 5 = 21.75 _____

5

x = 4.35 cmchallenge and extend

27. 6 __ w = 4.5 ___ 3

4.5w = 18

4.5w ____ 4.5

= 18 ___ 4.5

w = 4 m

4 __ y = 3 __ 6

3y = 24

3y

___ 3 = 24 ___

3

y = 8 m

10 ___ x = 8 __ 6

8x = 60

8x ___ 8 = 60 ___

8

x = 7.5 m

28. b 2

29. Let A represent the area of rectangle B, w represent width, and P represent perimeter.

A ______ 30.195

= ( 2 __ 3 )

2

A ______ 30.195

= 4 __ 9

9A = 120.78

9A ___ 9 = 120.78 ______

9

A = 13.42 cm 2

A = ℓ · w

w = A ___ 6.1

w = 13.42 _____ 6.1

w = 2.2 cm

P = 2(ℓ + w)P = 2(6.1 + 2.2)P = 2(8.3)P = 16.6 cm


CS10_A1_MESK710372_C01.indd 31 3/30/11 10:31:29 PM

precision And AccurAcY

CheCk it OUt! 1a. An ounce is a smaller than a pound, so 17 oz.

is more precise.

b. A hundredth of a meter is smaller than a tenth of a meter, so 7.85 m is more precise.

c. A gram is smaller than a kilogram, so 6000 g is more precise.

2a. Scales A and B measure to the nearest tenth of an ounce. Scale C measures to the nearest hundreth of ounce. Scale C is the most precise because a hundreth of an ounce is smaller than a tenth of an ounce.

b. Scale A: |16.00 - 16.3| = 0.3 Scale B: |16.00 - 15.8| = 0.2 Scale C: |16.00 - 16.07| = 0.07 Because 0.07 < 0.2 < 0.3, Scale C is the most

accurate. 3. 5.25 oz - 0.25 oz = 5.0 oz

5.25 oz + 0.25 oz = 5.50 ozThe lacrosse balls must be between 5.0 and 5.50 oz. Ball C does not fall between the specified tolerance because its weight is greater than the maximum weight of 5.50 oz.

4a. 4.1(0.05) = 0.21 4.1 in. ± 0.21 in. 3.89 in. - 4.31 in.

b. 475(0.025) = 11.88 475 m ± 11.88 m 463.12 m-486.88 m

c. 85(0.005) = 0.43 85 mg ± 0.43 mg 84.57 mg-85.43 mg

think and disCUss

1. Precision has to do with how small the units of measure are. Accuracy has to do with how close the measured quantity is to the actual quantity.

2. Possible answer: the length of a handmade candlestick

3. Measurement

Precision:weighingchemicalsin a lab

Accuracy:weightlimits on abridge

Tolerance:manufacturingnuts and bolts


1. The answer is precise because precision relates to the smallest unit that a ruler can measure.

2. The answer is accurate because accuracy relates to how close the scale can measure an object’s true mass.

3. 4.3 mLA tenth of a milliliter is smaller than a milliliter, so 4.3 mL is more precise.

4. 6.8 mA tenth of a meter is smaller than a meter, so 6.8 m is more precise.

5. 2.37 mgA hundredth of a milligram is smaller than a tenth of milligram, so 2.37 mg is the most precise.

6. 6.5 lbA tenth of a pound is smaller than a pound, so 6.5 lb is the most precise.

7. 47.3 ftA tenth of a foot is smaller than a foot, so 47.3 ft is more precise.

8. 13.9 ozA tenth of an ounce is smaller than an ounce, so 13.9 oz is more precise.

9a. Scale 1 measures to the nearest hundreth of a gram. Scales 2, 3, 4, and 5 measure to the nearest tenth of a gram. Scale 1 is the most precise because a hundreth of a gram is smaller than a tenth of a gram.

b. Scale 1: |10.00 - 9.98| = 0.02 Scale 2: |10.00 - 9.9| = 0.1 Scale 3: |10.00 - 10.1| = 0.1 Scale 4: |10.00 - 10.3| = 0.3 Scale 5: |10.00 - 9.8| = 0.2 Because 0.02 < 0.1 < 0.2 < 0.3, Scale 1 is the

most accurate.

10. Jen’s, Bill’s and Sasha’s odometers measure to the nearest hundreth of a mile. Rasheed’s odometer measures to the nearest thousandth of a mile. Rasheed’s odometer is the most precise because a thousandth of a mile is smaller than a hundredth of a mile.Jen: |1.000 - 1.01| = 0.01Bill: |1.000 - 0.97| = 0.03Rasheed: |1.000 - 0.989| = 0.011Sasha: |1.000 - 1.02| = 0.02Because 0.01 < 0.011 < 0.02 < 0.03, Jen’s odometer is the most accurate.


1-10

CS10_A1_MESK710372_C01.indd 32 3/30/11 10:31:30 PM

11. 595.5 g - 28.5 g = 567.0 g595.5 g + 28.5 g = 624.0 gNo. Basketball 4 does not fall between the specified tolerance because 624.5 g is greater than the maximum allowable mass of 624.0 g.

12. 51.5 in. - 2.5 in. = 49.0 in.51.5 in. + 2.5 in. = 54.0 in.No; basketballs 3 and 4 do not fall between the specified tolerance because their bounce heights of 60.0 in. and 55.2 in. are greater than the maximum allowable bounce height, respectively.

13. 50(0.02) = 1.0050 lb ± 1.00 lb49.00 lb-51.00 lb

14. 100(0.005) = 0.5100 yd ± 0.5 yd99.5 yd-100.5 yd

15. 25(0.04) = 1.0025 cm ± 1.00 cm24.00 cm-26.00 cm

16. 400(0.06) = 24.00400 L ± 24.00376.00 L-424.00 L

17. 250(0.04) = 10.00250 mm ± 10.00 mm240.00 mm-260.00 mm

18. 70(0.03) = 2.1070 kg ± 2.1067.90 kg-72.10 kg


19. One milligram is equal to one thousandth of a gram, so 4337 mg is the more precise measurement.

20. One inch is smaller than one foot, so 122 in. is more precise.

21. One lb is smaller than one ton, so 11,000 lb is more precise.

22. One cup is smaller than one pint, so 3 cups is more precise.

23. 6.83 cm equals 68.3 mm. A tenth of a millimeter is smaller than a millimeter, so 6.83 cm is more precise.

24. One tenth of a km is smaller than one mile, so 4.5 km is more precise.

25. 0.0127 m equals 1.27 cm. One hundredth of a centimeter is smaller than one centimeter, so 0.0127 m is more precise.

26. 115 oz equals 7.1875 lb. One ten thousandth of a pound is smaller than one hundreth of a pound, so 115 oz is more precise.

27a. Lucy’s cell phone measures time to the nearest tenth of a second. Juan and Pei’s cellphones measure time to the nearest hundredth of a second. Chandra’s cell phone measures time to the nearest thousandth of a second. Chandra’s cell phone is the most precise because a thousandth of a second is smaller than a hundredth of a second and a tenth of a second.

b. Lucy: |51.12 - 51.1| = 0.02 Juan: |51.12 - 52.23| = 0.11 Chandra: |51.12 - 51.769| = 0.649 Pei: |51.12 - 50.97| = 0.15 Because 0.02 < 0.11 < 0.15 < 0.649, Lucy’s

cell phone is the most accurate.

28. 100 in. + 0.25 in. = 100.25 in. 100 in. - 0.25 in. = 99.75 in.

100 5 __ 16

= 100.3125

Board 4’s length does not fall between the specified tolerance because 100.3125 in. is greater than the maximum allowable board length.

29. 45(0.02) = 0.9045 lb ± 0.90 lb44.10 lb-45.90 lb

30. 3(0.05) = 0.153 m ± 0.15 m2.85 m-3.15 m

31. 37(0.015) = 0.5637° C ± 0.56° C36.44° C-37.56° C

32. 750(0.03) = 22.50750 kg ± 22.50 kg727.50 kg-772.50 kg

33. 30(0.04) = 1.2030 ft ± 1.20 ft28.80 ft-31.20 ft

34. 550(0.08) = 44.00550 mL ± 44.00 mL506.00 mL-594.00 mL

35. 0.2(0.05) = 0.010.2 cm ± 0.01 cm0.19 cm-0.21 cm

36. 0.25(0.1) = 0.0250.25 kg ± 0.0250.23 kg-0.28 kg

37. 5456 mi

38. 3.63 m

39. 120 ft

40. 62.3 cg

41. 5,721 mg = 5.721 kg

5.721 kg equals 6 kg when rounded to the nearest kilogram.


CS10_A1_MESK710372_C01.indd 33 3/30/11 10:31:31 PM

42. 0.4586 km = 458.6 m0.4586 km equals 459 m when rounded to the nearest meter.

43. 16.270 liters = 16,270 mLA tenth of a milliliter is smaller than a milliliter, so 16,453.2 mL is more precise.

44. 437 mm = 43.7 cm A tenth of a centimeter is smaller than a centimeter, so 437 mm is more precise.

45. 0.265 cm = 2.65 mmA hundredth of a millimeter is smaller than a millimeter, so 0.265 cm is more precise.

46. 5.20 kg = 5200 mgA tenth of a milligram is smaller than a milligram, so 5200.0 mg is more precise.

47. 55 yd = 165 ftOne foot is smaller than one yard, so 165 ft is more precise.

48. One tenth of an hour equals six minutes. One minute is smaller than one tenth of an hour, so 67 min is more precise

49. 33 mg = 0.033 g1 mg = 0.001 gOne milligram equals one thousandth of a gram, so the measurements have the same precision.

50. 42.7 cm = 427 mmA tenth of a millimeter is smaller than a millimeter, so 427.0 mm is more precise.

51. 0.475 L = 475 mLA tenth of a milliliter is smaller than a milliliter, so 475.0 mL is more precise.

52. ( 2 ____ 100

) · 100 = 2%

100 m ± 2%

53. ( 2 ___ 50

) · 100 = 4%

50 g ± 4%

54. ( 12 ____ 240

) · 100 = 5%

240 ft ± 5%

55. ( 15 ____ 750

) · 100 = 2%

750 kg ± 2%

56. ( 0.25 ____ 25

) · 100 = 1%

25 in. ± 1%

57. ( 8.5 ____ 425

) · 100 = 2%

425 lb ± 2%

58. ( 1.5 ___ 60

) · 100 = 2.5%

60 oz ± 2.5%

59. ( 5.25 ____ 175

) · 100 = 3%

175 km ± 3%

60. 6 in. + 5 in. = 11 in.

11 in. _____ 2 = 5.5 in.

6 in. - 5 in. = 1 in.

1 in. ____ 2 = 0.5 in.

Length tolerance: 5.5 in. ± 0.5 in.

3 1 __ 2 in. + 4 1 __

4 in. = 7 3 __

4 in.

4 1 __ 4 in. - 3 1 __

2 in. = 3 __

4 in.

3 __ 4 in. · 1 __

2 in. = 3 __

8 in.

7 3 __ 4 in. · 1 __

2 = 31 ___

4 in. · 1 __

2 = 31 ___

8 in. = 3 7 __

8 in.

Width tolerance: 3 7 __ 8 in. ± 3 __

8 in.

61. 730.56 mm - 6.5 mm = 724.06 mm730.56 mm + 6.5 mm = 737.06 mmThe circumference range is 724.06 mm-737.06 mm.538.5 g - 28.5 g = 510.0 g538.5 g + 28.5 g = 567.0 gThe mass range is 510.0 g-567.0 g.1358.5 mm - 63.5 mm = 1295.0 mm1358.5 mm + 63.5 mm = 1422.0 mmThe bounce height range is 1295.0 mm-1422.0 mmBasketball 4 meets all specified tolerances because its circumference is between the circumference range, its mass is between the mass range, and its bounce height is between the bounce height range.

62. Possible answer: Nearest 1 __ 2 -inch; the manufacturer’s

meas. is to the nearest in., so the actual meas. is

between 38 1 __ 2 in. and 39

1 __ 2 in. This means Linda’s

door must be at least 39 1 __ 2 in. wide.

63. No; the precision is too great because (1) the length of the cut boards cannot be known to a greater precision than the length of the orig. board; (2) if Yusuf measures the cut boards the same way he measured the orig. board, the precision will be the same; and (3) part of the length of the orig. board equal to the thickness of the blade used to cut it will be lost.


CS10_A1_MESK710372_C01.indd 34 3/30/11 10:31:32 PM

StandardiZed teSt prep

64. 0.9728 g = 972.8 mgRounding 972.8 mg to the nearest milligram equals 973 mg. C is the correct answer.

65. 1.4 mm - 0.02 mm = 1.38 mm1.4 mm + 0.02 mm = 1.42 mm1.378 mm is less than the minimum tolerance of 1.38 mm. J is the correct answer.

66. 0.475 L = 475 mL0.5 L = 500 mLA tenth of a milliliter is smaller than a milliliter, so 475.3 milliliters is the most precise. A is the correct answer.


67. 5.002 g - 5.000 g = 0.002 g

0.002 _____ 5.000

· 100 = 0.04%

68. 525.3(0.005) = 2.6525.3 mi - 2.6 mi = 522.7 mi525.3 mi + 2.6 mi = 527.9 mi522.7 mi-527.9 mi

69. 384,403(0.0002) = 77 km384,403 - 77 km = 384,326 km384,403 + 77 km = 384,480 km384,326 km-384,480 km

reAdY to go on? section b quiz

1. 2 __ 3 = x ___

18

36 = 3x

36 ___ 3 = 3x ___

3

12 = x12 laptop models were sold.

2. 150 pages

_________ 5 h

· 1 h ______ 60 min

= 1 __ 2 page/min

3. 156 Calories ___________ 26 crackers

= 6 Calories/cracker

4. 1024 photographs

_______________ 8 h

= 128 photographs/h

5. -18 ____ n = 9 __ 2

-36 = 9n

-36 ____ 9 = 9n ___

9

-4 = n

6. d __ 5 = 2 __

4

4d = 10

4d ___ 4 = 10 ___

4

d = 2.5

7. 4 ___ 12

= r + 2 _____ 16

64 = 12r + 24 ____ - 24 _______ - 24 40 = 12r

40 ___ 12

= 12r ___ 12

10 ___ 3 = r

8. -3 ___ 7 = 6 _____

x + 6

-3x - 18 = 42 ________ + 18 ____ + 18 -3x = 60

-3x ____ -3

= 60 ___ -3

x = -20

9. RT ___ XZ

= ST ___ YZ

8 __ n = 12 ___ 9

72 = 12n

72 ___ 12

= 12n ____ 12

6 cm = n

10. AD ___ FJ

= CD ___ HJ

1.45 ____ 2.9

= n ___ 0.5

0.725 = 2.9n

0.725 _____ 2.9

= 2.9n ____ 2.9

0.25 yd = n

11. 2.5 ft; a tenth of a foot is the smaller measurement

12. 3 ft; a foot is the smaller measurement

13. 5910 g; a gram is the smaller measurement

14. 16.0 oz; a tenth of an ounce is the smaller measurement

15. 300(0.01) = 3 300 - 3 = 297 300 + 3 = 303 297 m - 303 m

16. 150(0.06) = 9 150 - 9 = 141 150 + 9 = 159 141 lb - 159 lb

17. 60(0.005) = 0.3 60 - 0.3 = 59.7 60 + 0.3 = 60.3 59.7 L - 60.3 L

18. 220(0.015) = 3.3 220 - 3.3 = 216.7 220 + 3.3 = 223.3 216.7 kg - 223.3 kg

studY guide: reView

VaRiables and eXpRessiOns

1. literal equation 2. ratio 3. ratio

4. 1.99 g 5. t + 3

6. qp = (1)(5) = 5 7. p ÷ q = (5) ÷ (1) = 5

8. q + p = (1) + (5) = 6

9. 150 ÷ m

150 ÷ (5) = 30

150 ÷ (6) = 25

150 ÷ (10) = 15

sOlVing eqUatiOns by adding OR sUbtRaCting

10. b - 16 = 20 ______ + 16 ____ + 16 b = 36

11. 4 + x = 2 _______ - 4 ___ - 4 x = -2

12. 9 + a = -12 _______ - 9 ____ - 9 a = -21

13. -7 + y = 11 _______ + 7 ___ + 7 y = 18

14. z - 1 __ 4 = 7 __

8

_____

+ 1 __ 4

____ + 1 __

4

z = 9 __ 8

15. w + 2 __ 3

= 3

_____

- 2 __ 3

____

- 3 __ 2

w = 7 __ 3

16. Let s represent the number of signatures Robin still needs.

27 + s = 108 ________ - 27 ____ - 27 s = 81 Robin still needs 81 signatures for her petition.


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1-2

CS10_A1_MESK710372_C01.indd 35 3/30/11 10:31:33 PM

sOlVing eqUatiOns by mUltiplying OR diViding

17. 35 = 5x

35 ___ 5 = 5x ___

5

7 = x

18. -3n = 10

-3n ____ -3

= 10 ___ -3

n = - 10 ___ 3

19. -30 = n __ 3

(3)(-30) = (3) ( n __ 3 )

-90 = n

20. x ___ -5

= -2.6

(-5) ( x ___ -5

) = (-5)(-2.6)

x = 13

21. 5y = 0

5y

___ 5 = 0 __

5

y = 0

22. -4.6r = 9.2

-4.6r _____ -4.6

= 9.2 _____ -4.6

r = -2

sOlVing twO-step and mUlti-step eqUatiOns

23. 4t - 13 = 57 ______ + 13 ____ + 13 4t = 70

4t __ 4 = 70 ___

4

t = 17.5

24. 5 - 2y = 15 ________ - 5 ___ - 5 -2y = 10

-2y

____ -2

= 10 ___ -2

y = -5

25. k __ 5 - 6 = 2

_____ + 6 ___ + 6

k __ 5 = 8

(5) k __ 5 = (5)8

k = 40

26. 5 __ 6 f - 3 __

4 f + 3 __

4 = 1 __

2

12 ( 5 __ 6 f - 3 __

4 f + 3 __

4 ) = 12 ( 1 __

2 )

10f - 9f + 9 = 6 f + 9 = 6 _____ - 9 ___ - 9 f = -3

27. 7x - 19x = 6 -12x = 6

-12x _____ -12

= 6 ____ -12

x = - 1 __ 2

28. 4 + 3a - 6 = 43 3a - 2 = 43 ______ + 2 ___ + 2 3a = 45

3a ___ 3 = 45 ___

3

a = 15

29. 8n + 22 = 70 _______ - 22 ____ - 22 8n = 48

8n ___ 8 = 48 ___

8

n = 6

3n = 3(6) = 18

30. 0 = 6n - 36 + 36 _______ + 36 36 = 6n

36 ___ 6 = 6n ___

6

6 = n

n - 5 = 6 - 5 = 1

31. 180 = 3a + 2a - 25 180 = 5a - 25 ____ + 25 _______ + 25 205 = 5a

205 ____ 5 = 5a ___

5

41 = a The angles are 3a = 3(41) = 123° and 2a - 25 = 2(41) - 25 = 82 - 25 = 57°.

sOlVing eqUatiOns with VaRiables On bOth sides

32. 4x + 2 = 3x ________ - 3x ____ - 3x x + 2 = 0 _____ - 2 ___ - 2 x = -2

33. -3r - 8 = -5r - 12 ________ + 5r _________ + 5r 2r - 8 = -12 _____ + 8 ____ + 8 2r = -4

2r __ 2 = -4 ___

2

r = -2

34. -a - 3 + 7 = 3a -a + 4 = 3a _______ + a ___ + a 4 = 4a

4 __ 4 = 4a ___

4

1 = a

35. -(x - 4) = 2x + 6 -x + 4 = 2x + 6 _______ + x _______ + x 4 = 3x + 6 ___ - 6 ______ - 6 -2 = 3x

-2 ___ 3 = 3x ___

3

- 2 __ 3 = x

36. 2 __ 3 n = 4n - 10 ___

3 n - 1 __

2

2 __ 3 n = 2 __

3 n - 1 __

2

_____

- 2 __ 3 n

_________ - 2 __

3 n

0 = - 1 __ 2 7

no solution

37. 0.2(7 + 2t) = 0.4t + 1.4 1.4 + 0.4t = 0.4t + 1.4 ________ - 0.4t __________ - 0.4t 1.4 = 1.4 3 all real numbers

38. Let p represent the number of photos. 0.36p = 2.52 + 0.08p _______ - 0.08p ___________ - 0.08p 0.28p = 2.52

0.28p

_____ 0.28

= 2.52 ____ 0.28

p = 9 Juan has 9 photos to print.


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CS10_A1_MESK710372_C01.indd 36 3/30/11 10:31:35 PM

sOlVing fOR a VaRiable

39. C = 360 ____ n

(n)C = (n) 360 ____ n

nC = 360

nC ___ C

= 360 ____ C

n = 360 ____ C

40. S = n __ 2 (a + ℓ)

( 2 __ n ) S = ( 2 __ n ) n __ 2 (a + ℓ)

2S ___ n = a + ℓ

___ - ℓ _____ - ℓ

2S ___ n - ℓ = a

41. a = d __ g

(g)a = (g) d __ g

ag = d

ag

___ a = d __ a

g = d __ a

g = 75 ____ 20.2

g ≈ 3.7 gal

sOlVing absOlUte-ValUe eqUatiOns

42. ⎜x + 6⎟ = 21 Case 1 x + 6 = 21 _____ -6 ___ -6 x = 15 x = 15, -27

Case 2

x + 6 = -21

_____ - 6 ___ -6

x = -27

43. 7 ⎜y - 5⎟ = 14

7 ⎜y - 5⎟

______ 7 = 14 ___

7

⎜y - 5⎟ = 2

Case 1

y - 5 = 2

+ 5 _____ y =

+ 5 ___

7

y = 7, 3

Case 2

y - 5 = -2

_____ +5 ___ +5 y = 3

44. 3 ⎜y⎟ + 4 = 31

_________ - 4 ___ -4

3 ⎜y⎟ = 27

3 ⎜y⎟

____ 3 = 27 ___

3

⎜y⎟ = 9

y = -9, 9

45. 12 = ⎜x - 5.4⎟

Case 1

x - 5.4 = 12.0

______ + 5.4 _____ + 5.4

x = 17.4

x = 17.4, -6.6

Case 2

x - 5.4 = -12.0

_______ + 5.4 _____ + 5.4

x = - 6.6

46. ⎜g + 6⎟ + 12 = 14

__________ - 12 ____ -12

⎜g + 6⎟ = 2

Case 1

g + 6 = 2

_____ -6 ___ -6

g = -4

g = -4, -8

Case 2

g + 6 = -2

_____ -6 ___ -6

g = -8

47. ⎜x⎟ = 5 __ 7

x = - 5 __ 7 , 5 __

7

48. ⎜x - 55⎟ = 5 Case 1 x - 55 = 5 ______ + 55 ____ +55 x = 60x = 60,50

Case 2

x - 55 = -5

______ + 55 ____ +55

x = 50

minimum: 50 mi/h maximum: 60 mi/h

Rates, RatiOs, and pROpORtiOns

49. Let x represent the number of cups of rice you will need.

x ___ 10

= 2 __ 6

6 · x = 2 · 10 6x = 20

6x ___ 6 = 20 ___

6

x = 3 1 __ 3

You will need 3 1 __ 3 cups of rice to make 10 servings of

casserole.

50. 30 cm ______ s · 1 m _______ 100 cm

· 3600 s _______ 1 h

= 1080 m/h

51. 75 ft ____ s · 1 mi ______ 5280 ft

· 60 s _____ 1 min

≈ 0.85 mi/min

52. n __ 8 = 2 ___

10

10n = 16

10n ____ 10

= 16 ___ 10

n = 1.6

53. 2 __ 9

= 12 ___ x

2x = 108

2x ___ 2

= 108 ____ 2

x = 54

54. 3 __ k = 9 ___

15

45 = 9k

45 ___ 9 = 9k ___

9

5 = k

55. 1 __ 3

= x _____ x - 6

x - 6 = 3x ______ - x ___ - x -6 = 2x

-6 ___ 2

= 2x ___ 2

-3 = x


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CS10_A1_MESK710372_C01.indd 37 3/30/11 10:31:36 PM

appliCatiOns Of pROpORtiOns

56. AC ___ DF

= BC ___ EF

x ____ 11.5

= 2 ___ 9.2

9.2x = 23

9.2x ____ 9.2

= 23 ___ 9.2

x = 2.5 cm

57. Let x represent the height of the tree.

x ___ 14

= 2 ____ 1.75

1.75x = 28

1.75x _____ 1.75

= 28 ____ 1.75

x = 16 The tree is 16 ft tall.

58. Original radius: 9 in. A = π r 2 = π(9 ) 2 = 81π

New radius: 9 · 2 __ 3 = 6 in.

A = π r 2 = π(6 ) 2 = 36π

Radii: Original

_______ New

= 9 __ 6 = 3 __

2

Area: Original

_______ New

= 81π ____ 36π

= 9 __ 4 = ( 3 __

2 )

2

The ratio of the areas is the square of the ratio of the radii.

pReCisiOn and aCCURaCy

59. 12 in.; an inch is the smaller measurement

60. 37.0 g; a tenth of a gram is the smaller measurement

61. 550 cm; a centimeter is the smaller measurement

62. 1.5 L; a tenth of a liter is the smaller measurement

63. 500(0.04) = 20 500 - 20 = 480 500 + 20 = 520 480 lb - 520 lb

64. 20(0.005) = 0.1 20 - 0.1 = 19.9 20 + 0.1 = 20.1 19.9 oz - 20.1 oz

65. 75(0.03) = 2.25 75 - 2.25 = 72.75 75 + 2.25 = 77.25 72.75 kg - 77.25 kg

66. 1035(0.1) = 103.5 1035 - 103.5 = 931.5 1035 + 103.5 = 1138.5 931.5 mm - 1138.5 mm

chApter test

1. c - a = (6) - (2) = 4

2. ab = (2)(3) = 6

3. c ÷ a = (6) ÷ (2) = 3

4. c __ b =

(6) ___

(3) = 2

5. b - a = (3) - (2) = 1

6. five less than n; the difference of n and 5

7. 8n; 8(5) = 40 mi

8. y - 7 = 2 ____ + 7 ___ + 7 y = 9

9. x + 12 = 19 _____ - 12 ____ - 12 x = 7

10. -5 + z = 8 _______ + 5 ___ + 5 z = 13

11. 9x = 72

9x ___ 9 = 72 ___

9

x = 8

12. m ___ -8

= -2.5

(-8) ( m ___ -8

) = (-8)(-2.5)

m = 20

13. 7 __ 8 a = 42

( 8 __ 7 ) 7 __

8 a = ( 8 __

7 ) 42

a = 48

14. 15 = 3 - 4x ___ - 3 ________ - 3 12 = -4x

12 ___ -4

= -4x ____ -4

-3 = x

15. 2a ___ 3 + 1 __

5 = 7 __

6

30 ( 2 __ 3 a + 1 __

5 ) = 30 ( 7 __

6 )

20a + 6 = 35 _______ - 6 ___ - 6 20a = 29

20a ____ 20

= 29 ___ 20

a = 1.45

16. 8 - (b - 2) = 11 8 - b + 2 = 11 10 - b = 11 ________ - 10 ____ - 10 -b = 1 (-1)(-b) = (-1)(1) b = -1

17. -2x + 4 = 5 - 3x ________ + 3x ______ + 3x x + 4 = 5 _____ - 4 ___ - 4 x = 1

18. 3(q - 2) + 2 = 5q - 7 - 2q 3q - 6 + 2 = 5q - 7 - 2q 3q - 4 = 3q - 7 ________ - 3q ________ - 3q -4 = -7 7 no solution

19. 5z = -3(z + 7) 5z = -3z - 21 ____ + 3z ________ + 3z 8z = -21

8z ___ 8 = -21 ____

8

z = -2.625

20. r - 2s = 14 ________ - r ___ - r -2s = 14 - r

-2s ____ -2

= 14 - r ______ -2

s = r - 14 ______ 2

21. V = 1 __ 3 bh

V = h __ 3 b

( 3 __ h ) V = ( 3 __

h ) h __

3 b

3V ___ h = b

22. P = 2(ℓ + w)

P __ 2 =

2(ℓ + w) _______

2

P __ 2 = ℓ + w

____ - w _____ - w

P __ 2 - w = ℓ

23. ⎜x - 14⎟ = 21 Case 1 x - 14 = 21 _______ + 14 ____ +14 x = 35 x = 35, -7

Case 2 x - 14 = - 21 _______ + 14 ____ +14 x = - 7


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CS10_A1_MESK710372_C01.indd 38 3/30/11 10:31:38 PM

24. 3 ⎜x⎟ + 5 = 8 _______ - 5 ___ -5 3 ⎜x⎟ = 3

3 ⎜x⎟

____ 3 = 3 __

3

⎜x⎟ = 1 x = 1, -1

25. ⎜2v ⎟ = 6 Case 1 2v = 6

2v ___ 2 = 6 __

2

v = 3

Case 2 2v = -6

2v ___ 2 = -6 ___

2

v = -3

26. 120 ____ 25

= 4.8 sheets/student

27. 1300 mg

________ day

· 1g ________

1000 mg ·

365 days ________

1 yr

13

102

73

= 13 · 73 ______ 2 g/yr = 949 ____

2 g/yr = 474.5 g/yr

28. 5 __ 4 = x ___

12

60 = 4x

60 ___ 4 = 4x ___

4

15 = x

29. 8 ___ 2z

= 15 ___ 60

480 = 30z

480 ____ 30

= 30z ____ 30

16 = z

30. x + 10 ______ 10

= 18 ___

12

12(x + 10) = 18(10) 12x + 120 = 180 _________ - 120 _____ - 120 12x = 60

12x ____ 12

= 60 ___ 12

x = 5

31. Let d represent the distance between the two cities on the map in inches.

1 ____ 500

= d ____ 875

875 = 500d

875 ____ 500

= 500d _____ 500

1.75 = d The cities are 1.75 in. apart on the map.

32. EF ___ RT

= EG ___ RS

6 __ x = 5 ___ 20

5x = 120 x = 24 in.

33. HJ ____ WX

= LK ___ ZY

3.84 ____ 3.2

= 2.04 ____ x

3.84x = 6.528

3.84x _____ 3.84

= 6.528 _____ 3.84

x = 1.7 m

34. 6.5 oz; a tenth of an ounce is the smaller measurement

35. 16 oz; an ounce is the smaller measurement 36. 3525 m; a meter is the smaller measurement

37. 25(0.01) = 0.25 25 - 0.25 = 24.75 25 + 0.25 = 25.25 24.75 ft–25.25 ft

38. 400(0.04) = 16 400 - 16 = 384 400 + 16 = 416 384 lb–416 lb

39. 250(0.005) = 1.25 250 - 1.25 = 248.75 250 + 1.25 = 251.25 248.75 cm–251.25 cm


CS10_A1_MESK710372_C01.indd 39 3/30/11 10:31:39 PM

chapter foundations of algebra 1 solutions key

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