· 20. 4x + 6y = 12 ___ - 6y _ - 6y 4x = 12 - 6y 4_ x 4 = 12 - 6y ___ 4 x = 3 - 3y 2 21. x r = n (r) x r = (r)n x= r n x n = r n x n = r 22

Solutions KeyEquations2

CHAPTER

ARE YOU READY? PAGE 69

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. 13�

25. Possible answer: a number plus 2 times itself

2-1 SOLVING ONE-STEP EQUATIONS, PAGES 72–77

CHECK IT OUT! PAGES 73–74

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

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

c. 6 + t = 14 ___________ - 6 - 6 t = 8

2a. p __

5 = 10

(5) ( p __

5 ) = (5)(10)

p = 50

b. 0.5y = -10(2)(0.5y) = (2)(-10) y = -20

c. c __ 8 = 7

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

c = 56

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. 1 __ 6 w = 102

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

w = 612

4. m __ 3

= 10,000

(3) m __ 3 = 3(10,000)

m = 30,000 ft

THINK AND DISCUSS, PAGE 75

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. 4, since (4) b __ 4 simplifies to b

3.

EXERCISES, PAGES 75–77GUIDED PRACTICE, PAGE 75

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

Copyright © by Holt, Rinehart and Winston. 21 Holt Algebra 1All rights reserved.

3660 a107_Chapter 2_021-058.indd 213660 a107_Chapter 2_021-058.indd 21 3/14/07 2:20:59 PM3/14/07 2:20:59 PM

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. t + 5 = -25 __________ - 5 - 5 t = -30

6. b + 2 __ 3 = 2

__________

- 2 __ 3 - 2 __

3

b = 4 __ 3

7. 4.2 = m + 3.6 ______________ - 3.6 - 3.6 0.6 = m

8. k __ 4 = 8

(4) k __ 4 = (4)8

k = 32

9. g _

1.9 = 10

(1.9) ( g _

1.9 ) = (1.9)(10)

g = 19

10. -2 = w _ -7

(-7)(-2) = (-7) ( w _ -7 )

14 = w

11. 4x = 28

4x _ 4 = 28 _

4

x = 7

12. 4m = 10

4m _ 4 = 10 _

4

m = 5 __ 2

13. -9j = -45

-9j

_ -9 = -45 _ -9

j = 5

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

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

16. -4.6 + q = 5 _______________ + 4.6 + 4.6 q = 9.6

17. 1 _ 2 d = 7

(2) ( 1 _ 2 d) = (2)(7)

d = 14

18. - 2 __ 3 + c = 2 __

3

+ 2 __ 3 + 2 __

3

c = 4 __ 3

19. 2 _ 3 s = -6

( 3 _ 2 ) ( 2 _

3 s) = ( 3 _

2 ) (-6)

s = -9

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

PRACTICE AND PROBLEM SOLVING, PAGES 75–77

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

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

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

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

25. b + 1 __ 2 = 1 __

2

_____

- 1 __ 2

____ - 1 __

2

b = 0

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

27. 4 2 __ 3 = r - 1 __

3

____

+ 1 __ 3

_____ + 1 __

3

5 = r

28. 2 = d + 1 __ 4

____

- 1 __ 4

______ - 1 __

4

7 __ 4 = d

29. x _ 2 = 12

(2) ( x _ 2 ) = (2)(12)

x = 24

30. 11 = -2z

11 _ -2 = -2z _ -2

- 11 ___ 2 = z

31. 5t = -15

5t _ 5 = -15 _

5

t = -3

32. 1.6 = d _ 3

(3)(1.6) = (3) ( d _ 3

)

4.8 = d

33. - j _ 6 = 6

(-6) (- j _ 6 ) = (-6)(6)

j = -36

34. -12 = -12u

-12 _ -12 = -12u _ -12

1 = u

35. -8.4 = -4n

-8.4 _ -4 = -4n _ -4

2.1 = n

36. h _ 8.1

= -4

(8.1) ( h _ 8.1

) = (8.1)(-4)

h = -32.4

37. -12 + f = 3 ________ + 12 ____ + 12 f = 15

38. -9 = -4 + g ___ + 4 _______ + 4 -5 = g

39. 4 _ 7 t = -2

( 7 _ 4 ) ( 4 _

7 t) = ( 7 _

4 ) (-2)

t = - 7 __ 2

40. - 4 _ 5 g = -12

(- 5 __ 4 ) (- 4 __

5 g) = (- 5 __

4 ) (-12)

g = 15

41. 26 = -4 + y ___ + 4 _______ + 4 30 = y

42. 5 _ 2 k = 5

( 2 _ 5 ) ( 5 _

2 k) = ( 2 _

5 ) (5)

k = 2

43. -9 = 3 _ 4 d

( 4 _ 3 ) (-9) = ( 4 _

3 ) ( 3 _

4 ) d

-12 = d

44. -5.2 + a = -8 ________ + 5.2 _____ + 5.2 a = -2.8

45. 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.

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



47. 5x = 45

5x ___ 5 = 45 ___

5

x = 9

48. x _ 3 = -8

(3) ( x _ 3 ) = (3)(-8)

x = -24

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

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

51. Let a represent the average depth. 30,246 = a + 17,366 ___________________ - 17,366 - 17,366 12,880 = a

The average depth of the Atlantic Ocean is 12,880 ft.

52. 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.

53. 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.

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

55a. 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.

c. Divide 780 by 7.

56. 63 + x = 90 ________ - 63 ____ - 63 x = 27

57. 42 + x = 90 ________ - 42 ____ - 42 x = 48

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

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

60. C; Since 10 _ 2 = 5, m = 10 is a solution for equation

C.

61a. 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.

CHALLENGE AND EXTEND, PAGE 77

62. 3 1 _ 5 + b = 4 _

5

________

- 3 1 __ 5

_____ - 3 1 __

5

b = - 12 _ 5

63. x - 7 _ 4

= 2 _ 3

_____

+ 7 _ 4

____

+ 7 __ 4

x = 29 _ 12

64. (1 1 _ 3 ) x = 2 2 _

3

4 _ 3 x = 8 _

3

( 3 _ 4 ) 4 _

3 x = ( 3 _

4 ) 8 _

3

x = 2

65. (3 1 _ 5

) b = 4 _ 5

16 _ 5

b = 4 _ 5

( 5 _ 16

) 16 _ 5

b = ( 5 _ 16

) 4 _ 5

b = 1 _ 4

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

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

67. 2p = 4

2p

_ 2 = 4 _

2

p = 2

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

68. 3x = 15

3x _ 3 = 15 _

3

x = 5

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

69. 2 + n = -11 _______ - 2 ____ - 2 n = -13

6n = 6(-13) = -78

SPIRAL STANDARDS REVIEW, PAGE 77

70. -63 ÷ (-7) = 9

71. 3 _ 7

÷ (- 4 _ 7

)

= 3 _ 7

· 7 _ -4

= - 3 _ 4

72. (-12)(-6) = 72

73. √ �� 225 = √ �� 15 2 = 15 m

74. √ �� 36 = √ � 6 2 = 6 ft 75. √ �� 100 = √ �� 10 2 = 10 cm

76. 11(104)= 11(100 + 4)= 11(100) + 11(4)= 1100 + 44 = 1144

77. 12(43)= 12(40 + 3)= 12(40) + 12(3)= 480 + 36 = 516

78. 3(46)= 3(40 + 6)= 3(40) + 3(6)= 120 + 18 = 138



2-2 SOLVING TWO-STEP EQUATIONS, PAGES 79–84


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

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. Let m represent Sara’s monthly fee. 15.95 + 12m = 735.95 _____________________ - 15.95 - 15.95 12m = 720

12m ____ 12

= 720 ____ 12

m = 60Sara’s monthly fee was $60.

b. Let m represent the number of records Lynda adds per month. 12 + 7m = 26 ________________ - 12 - 12 7m = 14

7m ___ 7 = 14 ___

7

m = 2Lynda adds 2 records each month.


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.


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. x __ 6 + 4 = 15

__________ - 4 - 4

x __ 6 = 11

(6) x __ 6 = (6)11

x = 66

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. 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

8. 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

9. x _ 8 - 1 _

2 = 6

_____

+ 1 _ 2

____ + 1 __

2

x _ 8 = 13 _

2

8 ( x _ 8 ) = 8 ( 13 _

2 )

x = 52

10. 1 __ 2 + 12x = 9 __

2

______________

- 1 __ 2 - 1 __

2

12x = 4 12x ____

12 = 4 ___

12

x = 1 __ 3



11. 5 __ 6 x - 1 __

3 = 5 __

2

___________

+ 1 __ 3 + 1 __

3

5 __ 6 x = 17 ___

6

( 6 __ 5 ) 5 __

6 x = ( 6 __

5 ) 17 ___

6

x = 17 ___ 5

12. 3 - 1 __ 2 r = 12

_________ - 3 - 3

- 1 __ 2 r = 9

(-2) (- 1 __ 2 r) = (-2)9

r = -18

13. 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.


14. 5 = 2g + 1 ___ - 1 ______ - 1 4 = 2g

4 _ 2 =

2g _

2

2 = g

15. 6h - 7 = 17 _____ + 7 ___ + 7 6h = 24

6h _ 6 = 24 _

6

h = 4

16. 15 = a __ 3 - 2

__________ + 2 + 2

17 = a __ 3

(3)17 = (3) a __ 3

51 = a

17. 3x + 3 = 18 _____ - 3 ___ - 3 3x = 15

3x _ 3 = 15 _

3

x = 5

18. 0.6g + 11 = 5 ________ - 11 ____ - 11 0.6g = -6

0.6g

_ 0.6

= -6 _ 0.6

g = -10

19. 32 = 5 - 3t ___ - 5 _______ - 5 27 = -3t

27 _ -3 = -3t _ -3

-9 = t

20. 2d + 1 _ 5 = 3 _

5

______

- 1 _ 5

____ - 1 __

5

2d = 2 _ 5

1 _ 2 (2d) = 1 _

2 ( 2 _

5 )

d = 1 _ 5

21. 1 = 2x + 1 _ 2

- 1 __ 2

______ - 1 _

2

1 _ 2 = 2x

1 _ 2 ( 1 _

2 ) = 1 _

2 (2x)

1 _ 4 = x

22. z _ 2 + 1 = 3 _

2

_____ - 1 ___ - 1

z _ 2 = 1 _

2

2 ( z _ 2 ) = 2 ( 1 _

2 )

z = 1

23. 2 _ 3

= 4j

_ 6

( 6 _ 4

) 2 _ 3

= ( 6 _ 4

) 4 _ 6

j

1 = j

24. 3 _ 4 = 3 _

8 x - 3 _

2

____

+ 3 __ 2

_______ + 3 _

2

9 _ 4 = 3 _

8 x

( 8 _ 3 ) 9 _

4 = ( 8 _

3 ) 3 _

8 x

6 = x

25. 1 _ 5

- x _ 5

= - 2 __ 5

________

- 1 __ 5

____

- 1 __ 5

- x __ 5

= - 3 __ 5

(-5) (- x __ 5

) = (-5) (- 3 __ 5

)

x = 3

26. 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.

27. 2n - 7 = 19 ______ + 7 ___ + 7 2n = 26

2n _ 2 = 26 _

2

n = 13

28. 8 - 3n = 2 - 8 - 8 -3n = -6

-3n _ -3 = -6 _ -3

n = 2

29. 30 + 5x = 80 _______________ - 30 - 30 5x = 50

5x ___ 5 = 50 ___

5

x = 10

30. 4x - 30 = 14 __________ ________ + 30 + 30 4x = 44

4x ___ 4

= 44 ___ 4

x = 11

31. 64 + 3x = -2 _______________ - 64 - 64 3x = -66

3x ___ 3 = -66 ____

3

x = -22

32. 6 + 2x = -8 _____________ - 6 - 6 2x = -14

2x ___ 2

= -14 ____ 2

x = -7

33a. 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 represent the number of scores. ns = 60 n(20) = 60

20n _ 20

= 60 _ 20

n = 3 There are 3 scores in 60 years.



34. 3t + 44 = 50 ______ - 44 ____ - 44 3t = 6

3t _ 3 = 6 _

3

t = 2

35. 3x - 6 = 18 ___________ + 6 + 6 3x = 24

3x ___ 3 = 24 ___

3

x = 8

36. 15 = c _ 3 - 2

___ + 2 _____ + 2

17 = c _ 3

3(17) = 3 ( c _ 3 )

51 = c

37. 2x + 6.5 = 15.5 _______ - 6.5 _____ - 6.5 2x = 9

2x _ 2 = 9 _

2

x = 4.5

38. 3.9w - 17.9 = -2.3 __________ + 17.9 ______ + 17.9 3.9w = 15.6

3.9w _ 3.9

= 15.6 _ 3.9

w = 4

39. 20 = x - 3x 20 = -2x

20 ___ -2 = -2x ____ -2

-10 = x

40. 5x + 9 = 39 _____ - 9 ___ - 9 5x = 30

5x _ 5 = 30 _

5

x = 6

41. 15 + 5.5m = 70 ___________ - 15 ____ - 15 5.5m = 55

5.5m _ 5.5

= 55 _ 5.5

m = 10

42. 7j + 3 = 24 __________ - 3 - 3 7j = 21

7j

__ 7 = 21 ___

7

j = 3

43. 3 __ 4 + x __

2 = 3

_______

- 3 __ 4

____ - 3 __

4

x __ 2 = 9 __

4

(2) x __ 2 = (2) 9 __

4

x = 9 __ 2 = 4.5

44. 50 = 3t - 4 ___ + 4 ______ + 4 54 = 3t

54 ___ 3 = 3t __

3

18 = t

45. 14.5 = 5.5n - 2 _____ + 2 ________ + 2 16.5 = 5.5n

16.5 ____ 5.5

= 5.5n ____ 5.5

3 = n

46. 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.

47. 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.

48. Let x represent the number of miles Karen traveled. 1.10 + 0.95x = 12.50 _____________ - 1.10 ______ - 1.10 0.95x = 11.40

0.95x _____ 0.95

= 11.40 _____ 0.95

x = 12Karen traveled 12 miles.

49a. 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

50. Possible answer: Subtract 2 from both sides or add m to both sides.

51. Possible answer: 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.

52. 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.

53. D;4m - 3 = 21 ______ + 3 ___ + 3 4m = 24

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

2s _ 2 = 54 _

2

s = 27




55. 11 ___ 2 + 3x = - 5 2 __

2

_________

- 11 ___ 2

_____ - 11 ___

2

3x =-18

3x ___ 3 = -18 ____

3

x = -6

56. 15 ___ 2 2

x - 15 = 33 ___ 2 2

________ + 15 ____ + 15

15 ___ 4 x = 93 ___

4

( 4 ___ 15

) 15 ___ 4 x = ( 4 ___

15 ) 93 ___

4

x = 93 ___ 15

= 31 ___ 5

57. -5.2x + 1.69 = -8.71 ___________ - 1.69 _______ - 1.69 -5.2x = -10.4

-5.2x _____ -5.2 = -10.4 ______ -5.2

x = 2

58. 1 __ 2 x - 12.75 = 21.25

__________ + 12.75 _______ + 12.75

1 __ 2 x = 34

(2) 1 __ 2 x = (2)34

x = 68

59. 169 = 37x - 4 2 _____ + 4 2 _______ + 4 2 185 = 37x

185 ____ 37

= 37x ____ 37

5 = x

60. 8.49 = 4.6x - 5.31 ______ + 5.31 __________ + 5.31 13.8 = 4.6x

13.8 ____ 4.6

= 4.6x ____ 4.6

3 = x

61a. 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


62. irrational

63. integer, rational, terminating decimal

64. repeating decimal, rational

65. terminating decimal, rational

66. 8(61) = 8(60) + 8(1) = 480 + 8 = 488

67. 9(28) = 9(20) + 9(8) = 180 + 72 = 252

68. 11(28) = 11(20) + 11(8) = 220 + 88 = 308

69. 13(21) = 13(20) + 13(1) = 260 + 13 = 273

70. 3(45)= 3(40 + 5)= 3(40) + 3(5)= 120 + 15 = 135

71. 7(19)= 7(20 - 1)= 7(20) - 7(1)= 140 - 7 = 133

72. 9(72)= 9(70 + 2)= 9(70) + 9(2)= 630 + 18 = 648

73. 8(33)= 8(30 + 3)= 8(30) + 8(3)= 240 + 24 = 264

74. 17 = k + 4 ___ - 4 _____ - 4 13 = k

75. x - 18 = 3 _____ + 18 ____ + 18 x = 21

76. a + 6 = -12 _____ - 6 ____ - 6 a = -18

77. -7 = q - 7 + 7 _____ + 7 0 = q

78. 12b = 60

12b ____ 12

= 60 ___ 12

b = 5

79. 7 = z __ 4

(4)7 = (4) z __ 4

28 = z

80. 3a = 24

3a ___ 3 = 24 ___

3

a = 8

81. t __ 6

= -7

(6) t __ 6

= (6)(-7)

t = -42

2-3 SOLVING MULTI-STEP EQUATIONS, PAGES 85–90


1a. 5m + 13 ________ 2 = 1

2 ( 5m + 13 ________ 2 ) = 2(1)

5m + 13 = 2 __________ - 13 ____ - 13 5m = -11

5m ___ 5 = -11 ____

5

m = - 11 ___ 5

b. 4 - 2z ______ 4

= -2

4 ( 4 - 2z ______ 4

) = 4(-2)

4 - 2z = -8 _________ - 4 ____ - 4 -2z = -12

-2z ____ -2 = -12 ____ -2

z = 6

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

-6a _ -6 = 5 _ -6

a = - 5 __ 6

b. -8 - 2d + 2 = 4-2d - 8 + 2 = 4 -2d - 6 = 4 ___________ + 6 ___ + 6 -2d = 10

-2d ____ -2 = 10 ___ -2

d = -5

c. 4x - 8 + 2x = 404x + 2x - 8 = 40 6x - 8 = 40 __________ + 8 ___ + 8 6x = 48

6x ___ 6 = 48 ___

6

x = 8



3a. 3(a + 1) - 4 = 53(a) + 3(1) - 4 = 5 3a + 3 - 4 = 5 3a - 1 = 5 _____________ + 1 ___ + 1 3a = 6

3a ___ 3 = 6 __

3

a = 2

b. -4(2 - y) = 8-4(2) - 4(-y) = 8 -8 + 4y = 8 ____________ + 8 ___ + 8 4y = 16

4y

___ 4 = 16 ___

4

y = 4

c. d + 3(d - 4) = 20d + 3(d) + 3(-4) = 20 d + 3d - 12 = 20 4d - 12 = 20 ______________ + 12 ____ + 12 4d = 32

4d ___ 4 = 32 ___

4

d = 8

4a. Let x represent monthly membership fee. 2(59.95 + 12m) = 1319.902(59.95) + 2(12m) = 1319.90 119.90 + 24m = 1319.90 ________________ - 119.90 ________ - 119.90 24m = 1200

24m ____ 24

= 1200 _____ 24

m = 50The monthly fee is $50.

4b. Let x represent the cost of the class. 5(c + 7) = 125.755(c) + 5(7) = 125.75 5c + 35 = 125.75 _________ - 35 ____ - 35 5c = 90.75

5c ___ 5 = 90.75 _____

5

c = 18.15The class costs $18.15.


1. Possible answer: To solve 3(z + 12) = 16, first distribute 3 on the left side.

2. Possible answer: First undo addition/subtraction, and then undo multiplication/division.

3.

EXERCISES, PAGES 88–90

GUIDED PRACTICE, PAGE 88

1. d + 3 _____ 4 = 2

4 ( d + 3 _____ 4 ) = 4(2)

d + 3 = 8 ________ - 3 ___ - 3 d = 5

2. 3x - 2 ______ 8 = 2

8 ( 3x - 2 ______ 8 ) = 8(2)

3x - 2 = 16 _________ + 2 ___ + 2 3x = 18

3x ___ 3 = 18 ___

3

x = 6

3. 6a - 1 ______ 7 = 1

7 ( 6a - 1 ______ 7 ) = 7(1)

6a - 1 = 7 _________ + 1 ___ + 1 6a = 8

6a ___ 6 = 8 __

6

a = 4 __ 3

4. n + 3 _____ 4 = 12

4 ( n + 3 _____ 4 ) = 4(12)

n + 3 = 48 ________ - 3 ___ - 3 n = 45

5. 2h - 5 ______ 2 = 0.5

2 ( 2h - 5 ______ 2 ) = 2(0.5)

2h - 5 = 1 _________ + 5 ___ + 5 2h = 6

2h ___ 2 = 6 __

2

h = 3

6. 10k + 9 _______ 4 = 5

4 ( 10k + 9 _______ 4 ) = 4(5)

10k + 9 = 20 __________ - 9 ___ - 9 10k = 11

10k ____ 10

= 11 ___ 10

k = 11 ___ 10

7. 9 - 2c + c = -13 9 - c = -13 _________ - 9 ___ - 9 -c = -22 (-1)(-c) = (-1)(-22) c = 22

8. 15y + 21 + 10 = 61 15y + 31 = 61 ____________ - 31 ____ - 31 15y = 30

15y

____ 15

= 30 ___ 15

y = 2

9. 8 = 3r - 5 + 4 8 = 3r - 1 ___ + 1 ______ + 1 9 = 3r

9 __ 3 = 3r __

3

3 = r

10. 42 = 4d - 6d + 6 42 = -2d + 6 ____ - 6 ________ - 6 36 = -2d

36 ___ -2 = -2d ____ -2

-18 = d



11. 2x + 0.3 - x = 3.32x - x + 0.3 = 3.3 x + 0.3 = 3.3 ___________ - 0.3 _____ - 0.3 x = 3

12. 2a + 3 + 2a = 112a + 2a + 3 = 11 4a + 3 = 11 __________ - 3 ___ - 3 4a = 8

4a ___ 4 = 8 __

4

a = 2

13. 3(x - 4) = 363(x) + 3(-4) = 36 3x - 12 = 36 ___________ + 12 ____ + 12 3x = 48

3x ___ 3 = 48 ___

3

x = 16

14. t(4 - 1) + 9 = 27t(4) + t(-1) + 9 = 27 4t - t + 9 = 27 3t + 9 = 27 _____________ - 9 ___ - 9 3t = 18

3t __ 3 = 18 ___

3

t = 6

15. 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

16. 17 = 4(a - 2) + 2a 17 = 4(a) + 4(-2) + 2a 17 = 4a - 8 + 2a 17 = 4a + 2a - 8 17 = 6a - 8 ___ + 8 _______ + 8 25 = 6a

25 ___ 6 = 6a ___

6

25 ___ 6 = a

17. 1 __ 2 (m - 6) = 12

1 __ 2 (m) + 1 __

2 (-6) = 12

1 __ 2 m - 3 = 12

____________ + 3 ___ + 3

1 __ 2 m = 15

2 ( 1 __ 2 m) = 2(15)

m = 30

18. 2 ( x __ 4 - 1) = 8

2 ( x __ 4 ) + 2(-1) = 8

x __ 2 - 2 = 8

+ 2 + 2

x __ 2 = 10

(2) x __ 2 = (2)10

x = 20

19. Let x represent the number of books on each bottom shelf.5 + 7 + 2x =24 12 + 2x = 24 _________ - 12 ____ - 12 2x = 12

2x ___ 2 = 12 ___

2

x = 6Each bottom shelf holds 6 books.


20. x - 1 _____ 2 = 5

2 ( x - 1 _____ 2 ) = 2(5)

x - 1 = 10 _______ + 1 ___ + 1 x = 11

21. 6 = 2w + 3 ______ 5

5(6) = 5 ( 2w + 3 ______ 5

)

30 = 2w + 3 ____ - 3 _______ - 3 27 = 2w

27 ___ 2

= 2w ___ 2

27 ___ 2

= w

22. -3 + y

______ 4 = 25

4 ( -3 + y

______ 4 ) = 4(25)

-3 + y = 100 _________ + 3 ___ + 3 y = 103

23. 5x - 2x + 3 = 24 3x + 3 = 24 __________ - 3 ___ - 3 3x = 21

3x ___ 3

= 21 ___ 3

x = 7

24. 11 = 2g + 6 - 5 11 = 2g + 1 ___ - 1 _______ - 1 10 = 2g

10 ___ 2 =

2g ___

2

5 = g

25. 2 + 0.5g + 9 = 612 + 9 + 0.5g = 61 11 + 0.5g = 61 ___________ - 11 ____ - 11 0.5g = 50 (2)0.5g = (2)50 g = 100

26. 4h - 7 + 2h = 74h + 2h - 7 = 7 6h - 7 = 7 __________ + 7 ___ + 7 6h = 14

6h ___ 6 = 14 ___

6

h = 7 __ 3

27. 34 = -5 - 6t + 3t 34 = -5 - 3t ___ + 5 ___ + 5 39 = -3t

39 ___ -3 = -3t ____ -3

-13 = t

28. 1.5v - 0.9v + 2.1 = 4.5 0.6v + 2.1 = 4.5 _______________ - 2.1 _____ - 2.1 0.6v = 2.4

0.6v ____ 0.6

= 2.4 ___ 0.6

v = 4

29. 3(d + 5) = 233(d) + 3(5) = 23 3d + 15 = 23 __________ - 15 ____ -15 3d = 8

3d ___ 3 = 8 __

3

d = 8 __ 3

30. 8(x + 2) = 328(x) + 8(2) = 32 8x + 16 = 32 _________ - 16 ____ - 16 8x = 16

8x ___ 8

= 16 ___ 8

x = 2



31. -12 = 5(k - 2) -12 = 5(k) + 5(-2) -12 = 5k - 10 ____ + 10 ________ + 10 -2 = 5k

-2 ___ 5 = 5k ___

5

- 2 __ 5 = k

32. 15 = 3(k - 6) 15 = 3(k) + 3(-6) 15 = 3k - 18 ____ + 18 ________ + 18 33 = 3k

33 ___ 3 = 3k ___

3

11 = k

33. 5 = 1 __ 2 (x - 6)

5 = 1 __ 2 (x) + 1 __

2 (-6)

5 = 1 __ 2 x - 3

___ + 3 _______ + 3

8 = 1 __ 2 x

(2)8 = (2) 1 __ 2 x

16 = x

34. 6(y - 4) = 06(y) + 6(-4) = 0 6y - 24 = 0 ___________ + 24 ____ + 24 6y = 24

6y

___ 6 = 24 ___

6

y = 4

35. Let c represent Casey’s portion.2c + c + 2 + 2 = 34 3c + 4 = 34 _____________ - 4 ___ - 4 3c = 30

3c ___ 3 = 30 ___

3

c =10Amanda: 2c = 2(10) = $20; Casey: c = $10

36. Let n be the number of brothers. 7.50(n + 2) = 307.50(n) + 7.50(2) = 30 7.50n + 15 = 30 ______________ - 15 ____ - 15 7.50n = 15

7.50n _____ 7.50

= 15 ____ 7.50

n = 2Marissa has 2 brothers.

37. 5(w + 10) = 455(w) + 5(10) = 45 5w + 50 = 45 ___________ - 50 ____ - 50 5w = -5

5w ___ 5 = -5 ___

5

w = -1

38. 9(p - 2) = 549(p) +9(-2) = 54 9p - 18 = 54 + 18 + 18 9p = 72

9p

___ 9 = 72 ___

9

p = 8

39. 1 __ 2 = 1 __

4 (y - 8)

1 __ 2 = 1 __

4 (y) + 1 __

4 (-8)

1 __ 2 = 1 __

4 y - 2

____ + 2 _______ + 2

5 __ 2 = 1 __

4 y

(4) 5 __ 2 = (4) 1 __

4 y

10 = y

40. 2x + 3 __ 4 + 6 __

4 = 16.75

2x + 2.25 = 16.75 __________ - 2.25 ______ - 2.25 2x = 14.5

2x ___ 2 = 14.5 ____

2

x = 7.25

41. 2.3w - 3.2 + 4.1 = -6 2.3w + 0.9 = -6 ______________ - 0.9 _____ - 0.9 2.3w = -6.9

2.3w ____ 2.3

= -6.9 _____ 2.3

w = -3

42. 19 = -4(h + 5) + h 19 = -4(h) + (-4)(5) + h 19 = -4h - 20 + h 19 = -4h + h - 20 19 = -3h - 20 ____ + 20 _________ + 20 39 = -3h

39 ___ -3 = -3h ____ -3

-13 = h

43. 3(2x - 4) + x __ 2 = 1

3(2x) + 3(-4) + x __ 2 = 1

6x - 12 + x __ 2 = 1

6x + x __ 2 - 12 = 1

13 ___ 2 x - 12 = 1

_______________ + 12 ____ + 12

13 ___ 2 x = 13

( 2 ___ 13

) 13 ___ 2 x = ( 2 ___

13 ) 13

x = 2

44. 2.5(4 - 2m) = 502.5(4) + 2.5(-2m) = 50 10 - 5m = 50 _______________ - 10 ____ - 10 -5m = 40

-5m _____ -5 = 40 ___ -5

m = -8

45. -4(d + 1) + 2(d - 2) = 10 ___ 3

-4(d) + (-4)(1) + 2(d) + 2(-2) = 10 ___ 3

-4d - 4 + 2d - 4 = 10 ___ 3

-4d + 2d - 4 - 4 = 10 ___ 3

-2d - 8 = 10 ___ 3

__________________________ + 8 ___ + 8

-2d = 34 ___ 3

(- 1 __ 2 ) (-2d) = (- 1 __

2 ) 34 ___

3

d = - 17 ___ 3



46. Let x represent the number. 6(3 + x) = 366(3) + 6(x) = 36 18 + 6x = 36 _________ - 18 ____ - 18 6x = 18

6x ___ 6 = 18 ___

6

x = 3The number is 3.

47. Let x represent the number.16 + 7 - 4x = 3 23 - 4x = 3 __________ - 23 ____ - 23 -4x = -20

-4x ____ -4 = -20 ____ -4


48. Let x represent the number. 3(x + 4) - x = 183(x) + 3(4) - x = 18 3x + 12 - x = 18 3x - x + 12 = 18 2x + 12 = 18 _____________ - 12 ____ - 12 2x = 6

2x ___ 2 = 6 __

2


49. Let x represent the number.

1 __ 2 x + 2(x - 5) = 0

1 __ 2 x + 2(x) + 2(-5) = 0

1 __ 2 x + 2x - 10 = 0

5 __ 2 x - 10 = 0

_______________ + 10 ____ + 10

5 __ 2 x = 10

( 2 __ 5 ) 5 __

2 x = ( 2 __

5 ) 10


50. (2x + 7) + 30 + 63 = 180 2x + 100 = 180 ________ - 100 _____ - 100 2x = 80

2x _ 2 = 80 _

2

x = 40

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

2x _ 2 = 65 _

2

x = 32.5

52. (4x - 80) + 60 + 60 = 180 4x + 40 = 180 _______ - 40 - 40 4x = 140

4x _ 4 = 140 _

4

x = 35

53. x + 3 - 2x + 5 = 10x - 2x + 3 + 5 = 10 -x + 8 = 10 _____________ - 8 ___ - 8 -x = 2 (-1)(-x) = (-1)2 x = -2

54. 9x - 5 - 6x - 1.3 = 09x - 6x - 5 - 1.3 = 0 3x - 6.3 = 0 _______________ + 6.3 _____ + 6.3 3x = 6.3

3x ___ 3 = 6.3 ___

3

x = 2.1

55. -5 = 7g + 5 - 6g -5 = 7g - 6g + 5 -5 = g + 5 ____ + 5 ______ + 5 -10 = g

56. 3(x - 2) - 5(2x + 1) = 33(x) + 3(-2) - 5(2x) + (- 5)(1) = 3 3x - 6 - 10x - 5 = 3 3x - 10x - 6 - 5 = 3 -7x - 11 = 3 __________________________ + 11 ____ + 11 -7x = 14

-7x ____ -7 = 14 ___ -7

x = -2

57. 5(2y - 7) + 8(y + 6) = 315(2y) + 5(-7) + 8(y) + 8(6) = 31 10y - 35 + 8y + 48 = 31 10y + 8y - 35 + 48 = 31 18y + 13 = 31 _______________________ - 13 ____ - 13 18y = 18

18y

____ 18

= 18 ___ 18

y = 1

58. 4x - 3(6 + x) - 1 = 24x - 3(6) + (-3)(x) - 1 = 2 4x - 18 - 3x - 1 = 2 4x - 3x - 18 - 1 = 2 x - 19 = 2 ____________________ + 19 ____ + 19 x = 21



59. 17 - (4 - 6z) + 4 = 4217 + (-1)(4) + (-1)(-6z) + 4 = 42 17 - 4 + 6z + 4 = 42 17 - 4 + 4 + 6z = 42 17 + 6z = 42 _________________________ - 17 ____ - 17 6z = 25

6z ___ 6 = 25 ___

6

z = 25 ___ 6

60. 42r - 2(13 - 5r) + 7 = 8542r - 2(13) + (-2)(-5r) + 7 = 85 42r - 26 + 10r + 7 = 85 42r + 10r - 26 + 7 = 85 52r - 19 = 85 ________________________ + 19 ____ + 19 52r = 104

52r ___ 52

= 104 ____ 52

r = 2

61. Let n represent the first number. (n) + (n + 1) = 5 2n + 1 = 5 ______ - 1 ___ - 1 2n = 4

2n _ 2

= 4 _ 2

n = 2 The numbers are n = 2 and n + 1 = 3.

62. 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.

63. 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.

64a. 4c + 2 + 5c = 92

b. 4c + 2 + 5c = 924c + 5c + 2 = 92 9c + 2 = 92 - 2 - 2 9c = 90

9c ___ 9 = 90 ___

9

c = 10

c. They removed 2 chairs from each row, and there were also 2 chairs left over to begin with. There are 9 rows, so 9(2) + 2 = 18 + 2 = 20 chairs were not used.

65. A is incorrect; possible answer: when using Comm. Prop. of Add., each sign should stay with its term.

66a. Days Acres

1 60

2 2(60) = 120

3 180

4 4(60) = 240

5 5(60) = 120

d d(60) = 60d

b. Let A represent the number of acres cleared.A = 60d

67. Dist. Prop., Comm. Prop. of Add., Comm. Prop. of Mult., Assoc. Prop. of Add.

68. 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.

69. D; 3(Josh’s comics + Howard’s comics) = 663( h + 12 + h ) = 66 3(h + h + 12) = 66 3(2h + 12) = 66

70. B; Step 2 should be 4g = 24

71. $10; let b represent bonus per show: 5(50 + b) = 3005(50) + 5(b) = 300 250 + 5b = 300 __________ - 250 _____ - 250 5b = 50

5b ___ 5 = 50 ___

5

b = 10


72. 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



73. 12 ( 1 __ 4 x - 1) = 12

12 ( 1 __ 4 x) + 12(-1) = 12

3x - 12 = 12 ______________ + 12 ____ + 12 3x = 24

3x ___ 3 = 24 ___

3

x = 8

74. (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

75. (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

76. 2(5 - y) - 5(y + 3) = -262(5) + 2(-y) - 5(y) - 5(3) = -26 10 - 2y - 5y - 15 = -26 -2y - 5y - 15 + 10 = -26 -7y - 5 = -26 ______________________ + 5 ___ + 5 -7y = -21

-7y ____ -7 = -21 ____ -7

y = 3

77. t(3 + 2) - 6(t - 5) - 22 = 6t(3) + t(2) - 6(t) - 6(-5) - 22 = 6 3t + 2t - 6t + 30 - 22 = 6 -t + 8 = 6 _________________________ - 8 ___ - 8 -t = -2 (-1)(-t) = (-1)(-2) t = 2

78a. Possible answer:

3x + 2(x + 6)

____________ 2(5 + 4)

= 14

5x + 12 _______ 18

= 14

5x + 12 = 252

b. An equivalent for the new equation would be 6x + 18 = 252, rather than 5x + 12 = 252. Since the last step would be to divide by 6 instead of 5, the solution would decrease.


79. Comm. Prop. of Add. 80. Comm. Prop. of Mult.

81. 5m + 3m= 8m

82. 22 c 2 - 14c(cannot combine terms)

83. 102v + 16v= 118v

84. 51b - b= 50b

85. 12c + x(cannot combine terms)

86. 1 __ 2

p + 1 __ 2

(cannot combine terms)

87. 10w + 4 = 34 _______ - 4 ___ - 4 10w = 30

10w ____ 10

= 30 ___ 10

w = 3

88. 11 = 3d - 4 ___ + 4 _______ + 4 15 = 3d

15 ___ 3

= 3d ___ 3

5 = d

89. 38 = -6d + 2 ___ - 2 ________ - 2 36 = -6d

36 ___ -6 = -6d ____ -6

-6 = d

TECHNOLOGY LAB: TRUE EQUATIONS, PAGE 91

TRY THIS, PAGE 91

1. sometimes 2. always

3. always 4. sometimes

5. sometimes 6. never

7. always 8. never

9. sometimes

2-4 SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES, PAGES 92–98


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 ✗ no solution

b. 2c + 7 + c = -14 + 3c + 21 3c + 7 = 3c + 7 ________ - 3c ________ - 3c 7 = 7 ✓ 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.


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.


1. The expressions are the same on both sides.

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. 1 + 3 __ 5 (q - 4) = 2 __

5 (q + 1)

1 + 3 __ 5 (q) + 3 __

5 (-4) = 2 __

5 (q) + 2 __

5 (1)

1 + 3 __ 5 q - 12 ___

5 = 2 __

5 q + 2 __

5

3 __ 5 q + 1 - 12 ___

5 = 2 __

5 q + 2 __

5

3 __ 5 q - 7 __

5 = 2 __

5 q + 2 __

5

_______________

- 2 __ 5 q

_____ - 2 __

5 q

1 __ 5 q - 7 __

5 = 2 __

5

_______________

+ 7 __ 5

________ + 7 __

5

1 __ 5 q = 9 __

5

(5) 1 __ 5 q = (5) 9 __

5

q = 9

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 ✓ all real numbers

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

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

13. 6 - 2x - 1 = 4x + 8 - 6x - 3 5 - 2x = 5 - 2x ______ + 2x ______ + 2x 5 = 5 ✓ 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

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 ✗ 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 ✓ all real numbers

31. -2(x + 2) = -2x + 1 -2(x) - 2(2) = -2x + 1 -2x - 4 = -2x + 1 ________ + 2x ________ + 2x -4 = 1 ✗ 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 ✓ 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. 5(x + 3) - 2 = 5x + 135(x) + 5(3) - 2 = 5x + 13 5x + 15 - 2 = 5x + 13 5x + 13 = 5x + 13always true

35. 2n + 1 = 2(n + 1) - 12n + 1 = 2(n) + 2(1) - 12n + 1 = 2n + 2 - 12n + 1 = 2n + 1always true

36. 3(k - 1) - 3(2k + 2) = 243(k) + 3(-1) - 3(2k) - 3(2) = 24 3k - 3 - 6k - 6 = 24 3k - 6k - 3 - 6 = 24 -3k - 9 = 24 + 9 + 9 -3k = 33

-3k ____ -3 = 33 ___ -3

k = -11

sometimes true (when k = -11)



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

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

9 x + 7 __

9

1 __ 3 (x) + 1 __

3 (1) = 2 __

9 x + 7 __

9

1 __ 3 x + 1 __

3 = 2 __

9 x + 7 __

9

__________

- 2 __ 9 x

_____ - 2 __

9 x

1 __ 9 x + 1 __

3 = 7 __

9

__________

- 1 __ 3

________ - 1 __

3

1 __ 9 x = 4 __

9

(9) 1 __ 9 x = (9) 4 __

9

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

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

b. 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 maintain its top speed for only 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.

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

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

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 ✗

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 ✓ 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.


72. 4x cm 73. 3y cm

74. undefined 75. 3(-21) = -63

76. 0 ÷ 7 _ 8

= 0 · 8 _ 7

= 0 · 8 _ 7

= 0 _ 7

= 0

77. 2 _ 5 ÷ 1 _

10

= 2 _ 5 · 10

= 2 · 10 _ 5

= 4

78. 5 ÷ (-5)

= 5 _ -5

= -1

79. -16 _ -8

= 16 _ 8

= 2

80. -1000 ÷ (-0.001)

= -1000 ÷ (- 1 ____ 100

)

= -1000 · -100 = 1,000,000

81. 500(-0.25)

= 500 (- 1 __ 4 )

= -500 _ 4

= -125

82. 4x - 44 = 8 _______ + 44 ____ + 44 4x = 52

4x _ 4 = 52 _

4

x = 13

83. 2(x - 3) = 24 2(x) + 2(-3) = 24 2x - 6 = 24 ______ + 6 ___ + 6 2x = 30

2x _ 2

= 30 _ 2

x = 15

84. -1 = x _ 4 - 3

___ + 3 ______ + 3 2 = x _

4

(4)(2) = (4) ( x _ 4 )

8 = x

85. 2x + 6 = 12 _____ - 6 ___ - 6 2x = 6

2x _ 2

= 6 _ 2

x = 3

CONCEPT CONNECTION, PAGE 100

1. 3850 _____ 14

= 275 acres per day

2. 275d = 9075

275d _____ 275

= 9075 _____ 275

d = 33 days

3. 5775 _____ 275

- 14 = d

21 - 14 = d 7 days = d

4. 1,440,000

_________ 21

≈ $68,571.43

READY TO GO ON? PAGE 101

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

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

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

4. 9 _ 8

= g + 1 _ 2

____

- 1 __ 2

______

- 1 _ 2

5 _ 8

= g

5. h _ 3 = -12

(3) ( h _ 3 ) = (3)(-12)

h = -36

6. -2.8 = w _ -3

(-3)(-2.8) = (-3) ( w _ -3 )

8.4 = w

7. 42 = 3c

42 _ 3 = 3c _

3

14 = c

8. -0.1b = 3.7

-0.1b _ -0.1 = 3.7 _ -0.1

b = -37

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. 2k + 15 = 29 _______ - 15 ____ - 15 2k = 14

2k ___ 2 = 14 ___

2

k = 7

11. -6a - 12 = 24 ________ + 12 ____ + 12 -6a = 36

-6a ____ -6 = 36 ___ -6

a = -6

12. 7 + 3b = -11 ________ - 7 ____ - 7 3b = -18

3b ___ 3 = -18 ____

3

b = -6

13. 1.6 = 0.4n - 2 _____ + 2 ________ + 2 3.6 = 0.4n

3.6 ___ 0.4

= 0.4n ____ 0.4

9 = n

14. 2r + 20 = 200 _______ - 20 ____ - 20 2r = 180

2r __ 2 = 180 ____

2

r = 90

15. 21 = 3g - 6 ___ + 6 _______ + 6 27 = 3g

27 ___ 3 =

3g ___

3

9 = g

16. 3 __ 5 k + 5 = 7

______ - 5 ___ - 5

3 __ 5 k = 2

( 5 __ 3 ) 3 __

5 k = ( 5 __

3 ) 2

k = 10 ___ 3

17. 2.5x + 4.5 = -8 _________ - 4.5 ______ - 4.5 2.5x = -12.5

2.5x ____ 2.5

= -12.5 ______ 2.5

x = -5

18. Let d represent the number of DVDs. 5.00 + 3.50d = 29.50 _____________ - 5.00 ______ - 5.00 3.50d = 24.50

3.50d _____ 3.50

= 24.50 _____ 3.50

d = 7Christine rented 7 DVDs.

19. 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.

20. 4(x - 7) = 2 4(x) + 4(-7) = 2 4x - 28 = 2 _______ + 28 ____ + 28 4x = 30

4x _ 4 = 30 _

4

x = 7.5

21. 2 __ 3 -

y __

4 = 5 ___

12

________

- 2 __ 3

____ - 2 __

3

- y __ 4 = - 1 __

4

(-4) (- y __ 4 ) = (-4) (- 1 __

4 )

y = 1

22. 5n + 6 - 3n = -12 2n + 6 = -12 ______ - 6 ____ - 6 2n = -18

2n _ 2 = -18 _

2

n = -9

23. 3 __ 4 (8 + 2y) = 3

3 __ 4 (8) + 3 __

4 (2y) = 3

6 + 3 __ 2 y = 3

___________ - 6 ___ - 6

3 __ 2 y = -3

2 __ 3 ( 3 __

2 y) = 2 __

3 (-3)

y = -2

24. 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.

25. 2(x + 3) = 242(x) + 2(3) = 24 2x + 6 = 24 _________ - 6 ___ - 6 2x = 18

2x ___ 2 = 18 ___

2

x = 9

x - 6 = 9 - 6 = 3

26. 4x - 3 = 2x + 5 ________ - 2x ________ - 2x 2x - 3 = 5 ______ + 3 ___ + 3 2x = 8

2x _ 2 = 8 _

2

x = 4

27. 3(2x - 5) = 2(3x - 2) 3(2x) + 3(-5) = 2(3x) + 2(-2) 6x - 15 = 6x - 4 _________ - 6x ________ - 6x -15 = -4 ✗ no solution

28. 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



29. 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

30. Let w represent the number of weeks. 700 - 35w = 450 + 15w _________ + 35w _________ + 35w 700 = 450 + 50w _____ - 450 ___________ - 450 250 = 50w

250 _ 50

= 50w _ 50

5 = w

450 + 15w = 450 + 15(5) = 450 + 75 = $525 Diego and Juan will both have $525 in their savings

accounts in 5 weeks.

2-5 SOLVING PROPORTIONS, PAGES 102–107


1. 18 ÷ 6 __ 5 = 18 × 5 __

6

= 15

2a. $52.50 ÷ 7 = $7.50/hour

b. 138 ÷ 23 = 6 envelopes/min

3a. -5 _ 2 =

y _

8

y = - 5 __ 2 × 8

y = -20

b. g + 3

_ 5

= 7 _ 4

g + 3 = 7 _ 4 × 5

g = 35 _ 4 - 12 _

4

g = 23 _ 4 = 5.75

4a. x _ 60

= 20 _ 100

100x = 1200 x = 12

b. 48 = 15% of x 48 = 0.15 · x 48 _ 0.15

= 0.15x _ 0.15

320 = x

5a. 1 ___ 18

= x __ 4

4 = 18x x ≈ 0.2 in.

b. 16 _ x = 32 _ 1

x = 16 _ 32

x = 0.5 ft = 6 in.


1. Multiply both sides by 4; use cross products.

2.


1. The ratios are equal.

2. Possible answer: Percent is a ratio to a total of 100.

3. x _ $64

= 3 _ 4

x = 3 _ 4 × $64

x = $48

4. x _ 341 trillion

= 2 _ 1

x = 2 _ 1

× 341 trillion

= 682 trillion

5. x _ 1 = 2000 _

40

x = 50 rotations/s

6. x _ 1

= 224,988

_ 12

x = 18,749 lb/cow

7. 3 _ z = 1 _ 8

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

8. x _ 3

= 1 _ 5

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

x = 3 _ 5

9. b _ 4 = 3 _

2

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

10. f + 3 _ 12

= 7 _ 2

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

11. -1 _ 5 = 3 _

2d

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

12. 3 _ 14

= s - 2 _ 21

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

13. x _ 40

= 75 _ 100

100x = 3000 x = 30

14. x = 12.5% of 168x = 0.125(168)x = 21

15. x = 115% of 57x = 1.15(57)x = 65.55

16. x = 70% of 8x = 0.7(8)x = 5.6

17. 25 _ 40

= x _ 100

40x = 2500

40x _ 40

= 2500 _ 40

x = 62.5%

18. 180 _ 225

= x _ 100

225x = 18,000

225x _ 225

= 18,000

_ 225

x = 80%

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. v _ 6 = 1 _

2

2(v) = 6(1)

2v _ 2 = 6 _

2

v = 3

25. 2 _ 5 = 4 _ y

2(y) = 4(5)

2y

_ 2 = 20 _

2

y = 10

26. 2 _ h = -5 _

6

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

(-5)h

_ (-5)

= 12 _ (-5)

h = -2.4

27. 3 _ 10

= b + 7 _ 20

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

10b _ 10

= -10 _ 10

b = -1

28. 60% of 80= 0.6(80)= 48

29. 35% of 90= 0.35(90)= 31.5

30. 1 _ 2 % of 500

= 0.005(500)= 2.5

31. 210% of 30= 2.1(30)= 63

32. 13 _ 52

= x _ 100

52x = 1300

52x _ 52

= 1300 _ 52

x = 25%

33. 27 _ 9 = x _

100

9x = 2700

9x _ 9 = 2700 _

9

x = 300%

34. 0.3 mm _ 3 cm

= 0.3 mm _ 3 cm

· 1 cm _ 10 mm

= 0.3 mm _ 30 mm

= 1 _ 100

So, the scale must be 1:100.

35. 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

36. m _ 3 = m + 4 _

7

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

4m _ 4 = 12 _

4

m = 3

37. 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

38. a _ 2 = a - 4 _

30

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

28a _ 28

= -8 _ 28

a = - 2 __ 7

39. 3 _ 2y

= 16 _ y + 2

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

6 _ 29

= 29y

_ 29

y = 6 _ 29

40. 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

41. 138,053,563

___ (138,053,563 + 143,368,343)

= 138,053,563 __

281,421,906

≈ 0.49056 ≈ 49% are male

143,368,343

___ (138,053,563 + 143,368,343)

= 143,368,343 __

281,421,906

≈ 0.50944 ≈ 51% are female

42. Possible answer: 1 _ 2 = 3 _

6 ; 1 _

4 = 5 _

20 ; 3 _

7 = 6 _

14 ;

they are proportions because their cross products are equal;

1 _ 2 ≠ 5 _

6 ; 1 _

2 ≠ 6 _

7 ; 3 _

4 ≠ 2 _

3 ;

they are not proportions because their cross products are not equal.

43a. 320 _______________________ 320 + 200 + 160 + 96 + 24

= 320 ____ 800

= 0.4 = 40%

b. Action:

200 ____ 800

= 0.25 = 25%

c. 24 ____ 800

= 0.03 = 3%

d. 320 - 25 ________ 800

= 295 ____ 800

= 0.36875 ≈ 36.9%



44a. 4 __ 3 = x ____

297

4(297) = x(3) 1188 = 3x

1188 _____ 3 = 3x ___

3

396 = xHigher frequency is 396 Hz.

b. 4 __ 3 = x ____

396

4(396) = x(3) 1584 = 3x

1584 _____ 3 = 3x ___

3

528 = xNew frequency is 528 Hz.

45. 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.

46. A;

x _____ 2500

= 1 ____ 100

x(100) = 1(2500) 100x = 2500

100x _____ 100

= 2500 _____ 100

x = $25

47. D; 14 ____ 100

= x ___ 60

48. 20;

x ____ 100

= 1 _ 2 (14)

_____ 35

x(35) = 1 __ 2 (14)(100)

35x = 700

35x ____ 35

= 700 ____ 35

x = 20


49. x _ 90 - x

= 4 _ 5

5(x) = 4(90 - x) 5x = 360 - 4x ______ + 4x ________ + 4x 9x = 360

9x _ 9 = 360 _

9

x = 40

One angle is 40°. Since the other angle is complementary to 40°, the other angle is 50°.

50. 672.2 people/ km 2 · 1 km 2 __ 1,000,000 m 2

= 0.0006722 people/ m 2

51. 2.75 _ 16

= x

16x = 2.75

16x _ 16

= 2.75 _ 16

x ≈ 0.172 ≈ 17.2%

52. 22 _ x = 73.5 _ 100

73.5x = 2200

73.5x _ 73.5

= 2200 _ 73.5

x ≈ 29.9

53. 3 __ x = 5 _____ x - 1

(x)3

____ x = (x)5

_____ x - 1

3 = 5x _____ x - 1

3(x - 1) = 5x _____ x - 1

(x - 1)

3(x - 1) = 5x


54. 8 2 = 8 · 8 = 64

55. (-3) 3 = (-3)(-3)(-3) = -27

56. (-3) 2 = (-3)(-3) = 9

57. - ( 1 _ 2 )

5 = - ( 1 5 _

2 5 ) = - ( 1 _

2 5 ) = - ( 1 _

32 ) = - 1 _

32

58. 6 3 59. 10 2

60. y 3 61. 2x - 12 = 5x + 3 _________ - 5x + 12 _________ - 5x + 12 -3x = 15

-3x _ -3 = 15 _ -3

x = -5

62. 3a - 4 = 6 - 7a ________ + 7a + 4 ________ + 4 + 7a 10a = 10

10a _ 10

= 10 _ 10

a = 1

63. 3x - 4 = 2x + 4 ________ - 2x + 4 ________ - 2x + 4 x = 8

2-6 SOLVING LITERAL EQUATIONS FOR A VARIABLE, PAGES 109–113


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

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




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 2� from both sides and then dividing both sides by 2.

3.


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 = �wh

V _ �h

= �wh _ �h

V _ �h

= 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

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. s = fw

s __ w = fw __ w

s __ w = f

b. f = 340 _____ 0.773

f ≈ 440 Hz c. The frequency decreases. As the wavelength

increases, the speed of sound is divided by greater and greater numbers, so the quotient decreases.

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

36. D; 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 needs 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)

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 )

500 (t - 1 __ 2 ) = d



42. s = 1 _ 2 g t 2

s = g t 2

___ 2

( 2 __ t 2

) (s) = ( 2 __ t 2

) ( g t 2

___ 2 )

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


46. 1 _ 4 (40) = $10 47. 2 _

5 (30) = 12 girls

48. 3 + 2 · x + 4 = 3 + 2(3) + 4 = 3 + 6 + 4 = 13

49. 24 ÷ 4 - x = 24 ÷ 4 - 12 = 6 - 12 = -6

50. 43 - 62 + x = 43 - 62 + 15 = -4

51. 18 = -2 + w ___ + 2 _______ + 2 20 = w

52. 2 = -3 + c ___ + 3 _______ + 3 5 = c

53. -8 + k = 4 _______ + 8 ___ + 8 k = 12

54. -15 + a = -27 ________ + 15 ____ + 15 a = -12

2-7 SOLVING ABSOLUTE-VALUE EQUATIONS, PAGES 114–119


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 ✓ 4 4 ✓

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 - 2.5 888

10.5 - 2.5 8 8 ✓

8 8 8

-5.5 - 2.5 -8 8 ✓

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

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 1x - 134 = 0.18 _______ + 134 _____ + 134 x = 134.18

Case 2x - 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.


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

2.




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

Check ________ |x| = 6 _________ |x| = 6 |6| 6 |-6| 6 6 6 ✓ 6 6 ✓

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 ✓ 9 9 ✓

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 ✓ 8 8 ✓

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 ✓ 18 18 ✓

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 __ 2 + 1 __

2

1 1

1

1 1 ✓

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

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

Case 2x - 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 ✓

-5 - 3 - 6 8 - 6 8 - 6 2

2 2 2 2 ✓

7. -8 = |x| ✗ no solution

8. |x| = 0 x = 0

Check ________ |x| = 0 |0| 0 0 0 ✓

9. |x + 4| = -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 ✓



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 ✓

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

5|x + 7|

_ 5 = -6 _

5

|x + 7| = - 6 _ 5 ✗

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 ✓

- 1 __ 5

1 __ 5

1 __ 5

1 __ 5 ✓

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 ✓ 22 22 ✓

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 ✓ 18 18 ✓

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 ✓ -4 -4 ✓

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 ✓ 18 18 ✓

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 ✓ 23.24 23.24 ✓



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 ✓ 2 _

3 2 _

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 ✓

_________________ |3x + 1| = 13

3 (- 14 _ 3 ) + 1 13

|-14 + 1| 13 |-13| 13 13 13 ✓

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 ✓

-2(4.4) + 3 -8.8 + 3 -5.8 5.8

5.8 5.8 5.8 5.8 ✓

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 ✓

24. 8 = 7 - |x| ___ -7 _______ -7 1 = -|x| (-1)(1) = (-1)(-|x|) - 1 = |x| ✗ 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 ✓

26. x - 3 + 14 = 5 ___________ - 14 ____ - 14 x - 3 = -9 ✗no 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 ✓

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 ✓

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.



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

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

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

Case 2x - 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. � - 65.1 = 0.2Case 1� - 65.1 = 0.2 _______ + 65.1 ______ + 65.1 � = 65.3

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

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

35. 3 - (-3)

________ 2 = 6 __

2 = 3

3 - 3 = 0Equation is x - 0 = 3 x = 3

36. 1 - (-1)

________ 2 = 2 __

2 = 1

1 - 1 = 0Equation is x - 0 = 1 x = 1

37. 5 - (-1)

________ 2 = 6 __

2 = 3

5 - 3 = 2Equation is x - 2 = 3

38. 2 - (-4)

________ 2 = 6 __

2 = 3

2 - 3 = -1Equation 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. f - 1666 b. f - 1666 = 1470

c. Case 1f - 1666 = 1470 ________ + 1666 ______ + 1666 f = 3136

Case 2f - 1666 = -1470 ________ + 1666 ______ + 1666 f = 196

Least frequency is 196 Hz; greatest frequency is 3136 Hz.

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.

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

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

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




50. P = 2� + 2w100 = 2 2x - 4 + 2x ________ - 2x _____________ - 2x 100 - 2x = 2 2x - 4 Case 1100 - 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 = xCase 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 = xThe 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 ✗no solution

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


53. 5 = p - 4.5 _____ + 4.5 _______ + 4.5 9.5 = p

54. -2 = y + 6 1 __ 2

_____

- 6 1 __ 2

_______ -6 1 __

2

-8 1 __ 2 = y

55. -12 + q = 3 ________ + 12 ____ + 12 q = 15

56. y - 4.3 = -5.7 ______ + 4.3 _____ + 4.3 y = -1.4

57. m __ 8 = 3 __

4

(8) m __ 8 = (8) 3 __

4

m = 6

58. 16 ___ y = 12 ___ 18

16(18) = 12(y) 288 = 12y

288 ____ 12

= 12y

____ 12

24 = y

59. 4 __ 5 = 12 _____

x + 6

4(x + 6) = 12(5)4(x) + 4(6) = 60 4x + 24 = 60 _________ - 24 ____ - 24 4x = 36

4x ___ 4 = 36 ___

4

x = 9

60. -2 ___ 3

= 5 ___ 2x

-2(2x) = 5(3) -4x = 15

-4x ____ -4 = 15 ___ -4

x = -3.75

61. m + 5n = 7 ______ - 5n ______ - 5n m = 7 - 5n

62. S = T + R ______ - R ______ - R S - R = T

63. 2y + 3x = 1 - 3x - 3x2y = 1 - 3x

2y

___ 2 = 1 - 3x ______

2

y = 1 - 3x ______ 2

64. 3 + w _____ z = x

3 + w _____ z (z) = x(z)

3 + w = xz _________ - 3 ______ - 3 w = xz - 3

65. c + d = 5 __ e

(c + d)e = ( 5 __ e ) e

(c + d)e = 5

(c + d)e

_______ (c + d)

= 5 ______ (c + d)

e = 5 ______ c + d

66. 6M - N = S _______ + N _______ + N 6M = S + N _______ - S ____ - S 6M - S = N

CONCEPT CONNECTION, PAGE 120

1. A;

352 ____ n = 4 __ 5

4n = 1760 n = 440

2. s = fw s __ f = w

w = 340 ____ 396

≈ 0.86 m/s



3. x - 2106.75 = 2079.25

Case 1x - 2106.75 = 2079.25 ________ +2106.75 ________ +2106.75 x = 4186 Hz

Case 2x - 2106.75 = -2079.25 ________ +2106.75 ________ +2106.75 x = 27.5 Hz

READY TO GO ON? PAGE 121

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. x = 40% of 25 x = 0.40(25) x = 10

10. x = 130% of 9 x = 1.30(9) x = 11.7

11. 35 = 70 · x 35 _

70 = 70x _

70

0.50 = x 50% = x

12. 640 = 400 · x 640 _

400 = 400x _

400

1.60 = x 160% = x

13. 16 = 80% of x 16 = 0.80x

16 _ 0.80

= 0.80x _ 0.80

20 = x

14. 28 = 200% of x 28 = 2x

28 _ 2 = 2x _

2

14 = x

15. Let t represent the total number of exhibits. 15 = 20% of t 15 = 0.20t

15 _ 0.20

= 0.20t _ 0.20

75 = t The total number of exhibits is 75.

16. 5j + s = t - 2 _________ + 2 _____ + 2 5j + s + 2 = t

17. h + p = 3(k - 8) h + p = 3(k) + 3(-8) h + p = 3k - 24 + 24 + 24 h + p + 24 = 3k

h + p + 24

__________ 3 = 3k ___

3

h + p

_____ 3 + 8 = k

18. 2x + 3y = 12 - 3y - 3y2x = 12 - 3y

2x ___ 2 =

12 - 3y _______

2

x = 6 - 3y

___ 2

19. x __ r = v

(r) x __ r = v(r)

x = vr

20. A = 1 __ 2 bh

2 __ b (A) = 2 __

b ( 1 __

2 bh)

2A ___ b = h

2(48)

_____ (12)

= h

8 cm = h

21. Case 1r = 7

Case 2r = -7

22. Case 1h + 4 = 11 _____ - 4 ___ - 4 h = 7

Case 2h + 4 = -11 _____ - 4 ____ - 4 h = -15

23. 2x + 4 = 0 2x + 4 = 0 - 4 - 4 2x = -4

2x ___ 2 = -4 ___

2

x = -2

24. 16 = 7 p + 3 + 30 ____ - 30 ____________ - 30

-14 = 7 p + 3 ∅

25. Let p represent the price of the car.p - 16,785 = 725Case 1p - 16,785 = 725 _________ + 16,785 ________ + 16,785 p = 17,510

Case 2p - 16,785 = -725 _________ + 16,785 ________ + 16,785 p = 16,060

The minimum guess is $16,060; the maximum guess is $17,510.



STUDY GUIDE: REVIEW, PAGES 122–125

1. literal equation 2. ratio

LESSON 2-1, PAGE 122

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

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

5. 9 + a = -12 _______ - 9 ____ - 9 a = -21

6. -7 + y = 11 _______ + 7 ___ + 7 y = 18

7. z - 1 _ 4 = 7 _

8

_____

+ 1 _ 4

____ + 1 __

4

z = 9 _ 8

8. w + 2 _ 3 = 3

_____

- 2 _ 3

____ - 2 __

3

w = 7 _ 3

9. 7.2 + t = 12.3 ________ - 7.2 _____ - 7.2 t = 5.1

10. 22.5 = x - 3 _____ + 3 _____ + 3 25.5 = x

11. 35 = 5x

35 ___ 5 = 5x ___

5

7 = x

12. -6s = 42

-6s ____ -6 = 42 ___ -6

s = -7

13. f ___ 15

= 8

(15) f ___ 15

= (15)8

f = 120

14. 3m = 45

3m ___ 3 = 45 ___

3

m = 15

15. 4j = 28

4j

__ 4 = 28 ___

4

j = 7

16. n __ 3 = 12

(3) n __ 3 = (3)12

n = 36

17. 4k = 3.5

4k ___ 4 = 3.5 ___

4

k = 0.875 or 7 __ 8

18. c __ 5 = 2.5

(5) c __ 5 = (5)2.5

c = 12.5

19. 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.

20. Let m represent the total amount of money.m = 6(35.50)m = $213Mrs. Wilson gave $213 in total.


21. 4t - 13 = 57 ______ + 13 ____ + 13 4t = 70

4t _ 4 = 70 _

4

t = 17.5

22. 5 - 2y = 15 ________ - 5 ___ - 5 -2y = 10

-2y

_ -2 = 10 _ -2

y = -5

23. 3z + 1 = 19 ______ - 1 ___ - 1 3z = 18

3z ___ 3 = 18 ___

3

z = 6

24. 18 = 30 - 3h ____ - 30 ____ - 30 -12 = -3h

-12 ____ -3 = -3h ____ -3

4 = h

25. k __ 5 - 6 = 2

_____ + 6 ___ + 6

k __ 5 = 8

(5) k __ 5 = (5)8

k = 40

26. 1 __ 6

f + 3 __ 4

= 1 __ 2

______

- 3 __ 4

____

- 3 __ 4

1 __ 6

f = - 1 __ 4

6 ( 1 __ 6

f) = 6 (- 1 __ 4

)

f = - 3 __ 2

27. 6h - 7 = 25 ______ + 7 ___ + 7 6h = 32

6h ___ 6 = 32 ___

6

h = 16 ___ 3

28. 17 = 29 - 4k ____ - 29 ____ - 29 -12 = -4k

-12 ____ -4 = -4k ____ -4

3 = k

29. 1 __ 4 + 3 __

4 t = 10

________

- 1 __ 4

____ - 1 __

4

3 __ 4 t = 39 ___

4

( 4 __ 3 ) 3 __

4 t = ( 4 __

3 ) 39 ___

4

t = 13

30. 18 - 4a = 34 _________ - 18 ____ - 18 -4a = 16

-4a ____ -4 = 16 ___ -4

a = -4

31. 0.2x + 0.5 = 0.9 _________ - 0.5 _____ - 0.5 0.2x = 0.4

0.2x ____ 0.2

= 0.4 ___ 0.2

x = 2

32. -0.6x + 1 = 0.4 _________ - 1 ___ - 1 -0.6x = -0.6

-0.6x _____ -0.6 = -0.6 _____ -0.6

x = 1

33. Let s represent Thomas’s score.s + s + (s + 5) = 281(s + s + s) + 5 = 281 3s + 5 = 281 _____________ - 5 ____ - 5 3s = 276

3s ___ 3 = 276 ____

3

s = 92




34. a - 12 + 2a = 27a + 2a - 12 = 27 3a - 12 = 27 __________ + 12 ____ + 12 3a = 39

3a ___ 3 = 39 ___

3

a = 13

35. 5y + 3 - 7y = 155y - 7y + 3 = 15 -2y + 3 = 15 - 3 - 3 -2y = 12

-2y ____ -2 = 12 ___ -2

y = -6

36. 4 + 3a - 6 = 43 3a - 2 = 43 ______ + 2 ___ + 2 3a = 45

3a _ 3 = 45 _

3

a = 15

37. 3(x + 2) = 243(x) + 3(2) = 24 3x + 6 = 24 _________ - 6 ___ - 6 3x = 18

3x ___ 3 = 18 ___

3

x = 6

38. h(5 - 2) + 8 = 17h(5) + h(-2) + 8 = 17 5h - 2h + 8 = 17 3h + 8 = 17 ______________ - 8 ___ - 8 3h = 9

3h ___ 3 = 9 __

3

h = 3

39. 8(z + 10) + z = 988(z) + 8(10) + z = 98 8z + 80 + z = 98 8z + z + 80 = 98 9z + 80 = 98 ______________ - 80 ____ - 80 9z = 18

9z ___ 9 = 18 ___

9

z = 2

40. -6 = 2.5(w - 2) -6 = 2.5(w) + 2.5(-2) -6 = 2.5w - 5 ___ + 5 ________ + 5 -1 = 2.5w

- 1 ___ 2.5

= 2.5w ____ 2.5

-0.4 or - 2 __ 5 = w

41. 0.3x - 1.1 - x = 0.30.3x - x - 1.1 = 0.3 -0.7x - 1.1 = 0.3 ____________ + 1.1 ____ + 1.1 -0.7x = 1.4

- 0.7x _____ -0.7 = 1.4 _____ -0.7

x = -2

42. 8n + 22 = 70 _______ - 22 ____ - 22 8n = 48

8n _ 8 = 48 _

8

n = 6

3n = 3(6) = 18

43. 0 = 6n - 36 + 36 _______ + 36 36 = 6n

36 _ 6 = 6n _

6

6 = n

n - 5 = 6 - 5 = 1

44. 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°. 45. Let a represent the cost of an adult ticket.

120(a - 3) + 80a = 640120(a) + 120(-3) + 80a = 640 120a - 360 + 80a = 640 120a + 80a - 360 = 640 200a - 360 = 640 ____________________ + 360 _____ + 360 200a = 1000

200a _____ 200

= 1000 _____ 200

a = $5The cost of an adult ticket is $5. The cost of a child ticket is $5 - $3 = $2.


46. 4x + 2 = 3x ________ - 3x ____ - 3x x + 2 = 0 _____ - 2 ___ - 2 x = -2

47. -3r - 8 = -5r - 12 ________ + 5r _________ + 5r 2r - 8 = -12 _____ + 8 ____ + 8 2r = -4

2r _ 2 = -4 _

2

r = -2

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

4 _ 4 = 4a _

4

1 = a

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

-2 _ 3 = 3x _

3

- 2 __ 3 = x



50. 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 ✗

no solution

51. 0.2(7 + 2t) = 0.4t + 1.4 1.4 + 0.4t = 0.4t + 1.4 ________ - 0.4t __________ - 0.4t 1.4 = 1.4 ✓ all real numbers

52. 0.5x - 1.7 = 0.3x - 1 _____ -0.3x _____ -0.3x 0.2x - 1.7 = -1 __________ +1.7 _________ +1.7 0.2x = 0.7

0.2x ____ 0.2

= 0.7 ___ 0.2

x = 3.5

53. -2c + 8 = 5c - 10 ________ + 2c ____ + 2c 8 = 7c - 10 ________ + 10 _________ + 10 18 = 7c

18 ___ 7 = 7c ___

7

18 ___ 7 = c

54. 7x - 28 = 3x _________ - 7x ____ - 7x -28 = -4x

-28 ____ -4 = -4x ____ -4

7 = x

55. 9x - 3 = 6x + 15 ________ - 6x ____ - 6x 3x - 3 = 15 ________ + 3 _________ + 3 3x = 18

3x ___ 3 = 18 ___

3

x = 6

56. 3 __ 4 n + 1 = 5 __

4 n + 2

_________

- 3 __ 4 n

____ - 3 __

4 n

1 = 1 __ 2 n + 2

________ - 2 ___ - 2

-1 = 1 __ 2 n

2(-1) = 2 ( 1 __ 2 n)

-2 = n

57. 1 __ 3 x + 2 __

3 = 5 __

3

______

- 2 __ 3

____ - 2 __

3

1 __ 3 x = 1

(3) 1 __ 3 x = (3)1

x = 3

58. 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 develop.

LESSON 2-5, PAGES 124–125

59. 320 _ 20

= 16 _ 1

60. n _ 8 = 2 _

10

10n = 16

10n _ 10

= 16 _ 10

n = 1.6

61. 2 _ 9

= 12 _ x

2x = 108

2x _ 2 = 108 _

2

x = 54

62. 1 ___ 30

= x ___ 41

30x = 41

30x ____ 30

= 41 ___ 30

x ≈ 1.37 ft

63. x = 2.3% of 230 x = 0.023(230) x = 5.29

64. x = 115% of 2700 x = 1.15(2700) x = 3105

65. 12 = 18x

12 _ 18

= 18x _ 18

2 _ 3 = x

0.667 ≈ x 66.7% ≈ x

66. 56 = 14x

56 _ 14

= 14x _ 14

4 = x 400% = x

67. 120 = 90% of x 120 = 0.90x

120 _ 0.90

= 0.90x _ 0.90

133.3 ≈ x

68. 90 = 37.5% of x 90 = 0.375x

90 _ 0.375

= 0.375x _ 0.375

240 = x

69. 32 _ 32 + 8

= 32 _ 40

= 4 _ 5 = 0.80 = 80%


70. C = 360 _ n

(n)C = (n) 360 _ n

nC = 360

nC _ C

= 360 _ C

n = 360 _ C

71. S = n _ 2

(a + �)

( 2 _ n ) S = ( 2 _ n ) n _ 2

(a + �)

2S _ n = a + �

___ - � _____ - �

2S _ n - � = a

72. 0.25x + y = 225 ________ - y ____ - y 0.25x = 225 - y

0.25x _ 0.25

= 225 - y

_ 0.25

x = 225 - y

_ 0.25

73. 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


74. Case 1x + 6 = 21 _____ - 6 ___ - 6 x = 15

Case 2x + 6 = -21 _____ - 6 ____ - 6 x = -27



75. Case 1 7(y - 5) = 147(y) + 7(-5) = 14 7y - 35 = 14 + 35 + 35 7y = 49

7y

___ 7 = 49 ___

7

y = 7

Case 2 7(y - 5) = -147(y) + 7(-5) = -14 7y - 35 = -14 + 35 + 35 7y = 21

7y

___ 7 = 21 ___

7

y = 3

76. 3 y + 4 = 31 ______ - 4 ___ - 4 3 y = 27

3 y

____ 3 = 27 ___

3

y = 9

Case 1y = 9

Case 2y = -9

77. Case 1 12 = x - 5.4 _____ + 5.4 _______ + 5.4 17.4 = x

Case 2-12 = x - 5.4 _____ + 5.4 _______ + 5.4 -6.6 = x

78. g + 6 + 12 = 14 ___________ - 12 ____ - 12 g + 6 = 2

Case 1g + 6 = 2 _____ - 6 ___ - 6 g = -4

Case 2g + 6 = -2 _____ - 6 ___ - 6 g = -8

79. Case 1

x = 5 __ 7

Case 2

x = - 5 __ 7

80. Let s represent Jason’s speed. s - 55 = 5

Case 1s - 55 = 5 ______ + 55 ____ + 55 s = 60

Case 2s - 55 = -5 ______ + 55 ____ + 55 s = 50

Maximum speed is 60 mi/h; minimum speed is 50 mi/h.

CHAPTER TEST, PAGE 126

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

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

3. -5 + z = 8 _______ + 5 ___ + 5 z = 13

4. 9x = 72

9x _ 9 = 72 _

9

x = 8

5. m _ -8 = -2.5

(-8) ( m _ -8 ) = (-8)(-2.5)

m = 20

6. 7 _ 8 a = 42

( 8 _ 7 ) 7 _

8 a = ( 8 _

7 ) 42

a = 48

7. 15 = 3 - 4x ___ - 3 ________ - 3 12 = -4x

12 _ -4 = -4x _ -4

-3 = x

8. 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

9. 8 - (b - 2) = 11 8 - b + 2 = 11 10 - b = 11 ________ - 10 ____ - 10 -b = 1 (-1)(-b) = (-1)(1) b = -1

10. -2x + 4 = 5 - 3x ________ + 3x ______ + 3x x + 4 = 5 _____ - 4 ___ - 4 x = 1

11. 3(q - 2) + 2 = 5q - 7 - 2q 3q - 6 + 2 = 5q - 7 - 2q 3q - 4 = 3q - 7 ________ - 3q ________ - 3q -4 = -7 ✗ no solution

12. 5z = -3(z + 7) 5z = -3z - 21 ____ + 3z ________ + 3z 8z = -21

8z _ 8 = -21 _

8

z = -2.625

13. m - 2.7 = -1.5m + 1 _________ - m ______ - m -2.7 = -2.5m + 1 _________ - 1 ___ - 1 -3.7 = -2.5m

-3.7 _____ -2.5 = -2.5m ______ -2.5

1.48 = m

14. x - 3.6 = 10.2 - 3.2x - 3.6 = 7 ______ + 3.6 _____ + 3.6 x = 10.6

15. c + 13.5 = 20 _______ - 13.5 ______ - 13.5 c = 6.5

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

-2s _ -2 = 14 - r _ -2

s = r - 14 _ 2

17. V = 1 _ 3 bh

V = h _ 3 b

( 3 _ h ) V = ( 3 _

h ) h _

3 b

3V _ h = b

18. P = 2(� + w)

P _ 2 =

2(� + w) _

2

P _ 2 = � + w

____ - w _____ - w

P _ 2 - w = �

19. 2x + a = 4 ______ - a _____ - a 2x = 4 - a

2x ___ 2 = 4 - a _____

2

x = 4 - a _____ 2



20. 4x + 6y = 12 _______ - 6y _______ - 6y 4x = 12 - 6y

4x ___ 4 =

12 - 6y _______

4

x = 3 - 3 __ 2 y

21. x __ r = n

(r) x __ r = (r)n

x = r n

x __ n = r n __ n

x __ n = r

22. Let b represent the number of blue marbles.

4 _ 7 = 16 _

b

4b = 112

4b _ 4 = 112 _

4

b = 28 There are 28 blue marbles in the bag.

23. 4.99 _ 3 ≈ $1.66/videotape

24. 120 _ 25

= 4.8 sheets/student

25. 5 _ 4 = x _

12

60 = 4x

60 _ 4 = 4x _

4

15 = x

26. 8 _ 2z

= 15 _ 60

480 = 30z

480 _ 30

= 30z _ 30

16 = z

27. x + 10 _ 10

= 18 _ 12

12(x + 10) = 18(10) 12x + 120 = 180 _________ - 120 _____ - 120 12x = 60

12x _ 12

= 60 _ 12

x = 5

28. x __ 8 = 1 __

4

(8) x __ 8 = (8) 1 __

4

x = 2

29. 5 ___ 12

= - 4 __ f

5(f) = -4(12) 5f = -48

5f __ 5 = -48 ____

5

f = -9.6

30. 3 ___ 10

= x + 1 _____ 15

(15) 3 ___ 10

= (15) x + 1 _____ 15

4.5 = x + 1 _______ - 1 _____ - 1 3.5 = x

31. c - 4 _____ 5 = -c ___

2

2(c - 4) = 5(-c)2(c) + 2(-4) = -5c 2c - 8 = -5c ________ - 2c ____ - 2c -8 = -7c

-8 ___ -7 = -7c ____ -7

8 __ 7 = c

32. 3n ___ 2 = 2 __

3

( 2 __ 3 ) 3n ___

2 = ( 2 __

3 ) 2 __

3

n = 4 __ 9

33. w __ 6 = 5 __

2

(6) w __ 6 = (6) 5 __

2

w = 15

34. 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.

35. 1 __ 8 = 0.125; 1 __

2 = 0.5; 20% = 0.2; 30% = 0.3

0.125 < 0.2 < 0.3 < 0.5 < 0.625 < 1

Order is 1 __ 8 , 20%, 30%, 1 __

2 , 0.625, 1

36. x = 23% of 46 x = 0.23(46) x = 10.58

37. 37.5 = 60% of x 37.5 = 0.60x

37.5 _ 0.60

= 0.60x _ 0.60

62.5 = x

38. 35 = 175x

35 _ 175

= 175x _ 175

1 _ 5 = x

0.20 = x 20% = x

39. x = 29% of 32x = 0.29(32)x = 9.28

40. 84.41 = 23% of x 84.41 = 0.23x

84.41 _____ 0.23

= 0.23x _____ 0.23

367 = x

41. Case 1x - 14 = 21 ______ + 14 ____ + 14 x = 35

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

42. 13 = y + 2 - 3 ___ + 3 __________ + 3 16 = y + 2 Case 1 16 = y + 2 ___ - 2 _____ - 2 14 = y

Case 2-16 = y + 2 ____ - 2 ______ - 2 -18 = y

43. Case 1 4z = 20

4z ___ 4 = 20 ___

4

z = 5

Case 2 4z = -20

4z ___ 4 = -20 ____

4

z = -5

44. 3 x + 5 = 8 _______ - 5 ___ - 5 3 x = 3Case 1 3x = 3

3x ___ 3

= 3 __ 3

x = 1

Case 2 3x = 3

3x ___ 3

= -3 ___ 3

x = -1



45. 3 g + 1 + 5 = 7 ___________ - 5 ___ - 5 3 g + 1 = 2

Case 1 3(g + 1) = 23(g) + 3(1) = 2 3g + 3 = 2 _________ - 3 ___ - 3 3g = -1

3g

___ 3 = -1 ___

3

g = - 1 __ 3

Case 2 3(g + 1) = -23(g) + 3(1) = -2 3g + 3 = -2 _________ - 3 ___ - 3 3g = -5

3g

___ 3 = -5 ___

3

g = - 5 __ 3

46. Case 1 2v = 6

2v ___ 2 = 6 __

2

v = 3

Case 2 2v = -6

2v ___ 2 = -6 ___

2

v = -3



Documents

· 20. 4x + 6y = 12 _______ - 6y _______ - 6y 4x = 12 - 6y 4___ x 4 = 12 - 6y _______ __ 4 x = 3 - __3y 2 21. x__ r = n (r) x__ r = (r)n x= r n __x n = r n x__ n = r 22

· 20. 4x + 6y = 12 ___ - 6y _ - 6y 4x = 12 - 6y 4_ x 4 = 12 - 6y ___ 4 x = 3 - 3y 2 21. x r = n (r) x r = (r)n x= r n x n = r n x n = r 22