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.
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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
Copyright © by Holt, Rinehart and Winston. 22 Holt Algebra 1All rights reserved.
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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
Copyright © by Holt, Rinehart and Winston. 23 Holt Algebra 1All rights reserved.
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2-2 SOLVING TWO-STEP EQUATIONS, PAGES 79–84
CHECK IT OUT! PAGES 79–81
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.
THINK AND DISCUSS, PAGE 82
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.
EXERCISES, PAGES 82–84GUIDED PRACTICE, PAGE 82
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
Copyright © by Holt, Rinehart and Winston. 24 Holt Algebra 1All rights reserved.
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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.
PRACTICE AND PROBLEM SOLVING, PAGES 82–84
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.
Copyright © by Holt, Rinehart and Winston. 25 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 253660 a107_Chapter 2_021-058.indd 25 3/14/07 2:21:09 PM3/14/07 2:21:09 PM
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
Copyright © by Holt, Rinehart and Winston. 26 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 263660 a107_Chapter 2_021-058.indd 26 3/14/07 2:21:11 PM3/14/07 2:21:11 PM
CHALLENGE AND EXTEND, PAGE 84
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
SPIRAL STANDARDS REVIEW, PAGE 84
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
CHECK IT OUT! PAGES 85–87
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
Copyright © by Holt, Rinehart and Winston. 27 Holt Algebra 1All rights reserved.
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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.
THINK AND DISCUSS, PAGE 87
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
Copyright © by Holt, Rinehart and Winston. 28 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 283660 a107_Chapter 2_021-058.indd 28 3/14/07 2:21:14 PM3/14/07 2:21:14 PM
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.
PRACTICE AND PROBLEM SOLVING, PAGES 88–90
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
Copyright © by Holt, Rinehart and Winston. 29 Holt Algebra 1All rights reserved.
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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
Copyright © by Holt, Rinehart and Winston. 30 Holt Algebra 1All rights reserved.
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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
x = 5The number is 5.
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
x = 3The number is 3.
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
x = 4The number is 4.
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
Copyright © by Holt, Rinehart and Winston. 31 Holt Algebra 1All rights reserved.
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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
CHALLENGE AND EXTEND, PAGE 90
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
Copyright © by Holt, Rinehart and Winston. 32 Holt Algebra 1All rights reserved.
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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.
SPIRAL STANDARDS REVIEW, PAGE 90
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
CHECK IT OUT! PAGES 92–94
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
Copyright © by Holt, Rinehart and Winston. 33 Holt Algebra 1All rights reserved.
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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.
THINK AND DISCUSS, PAGE 95
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.
EXERCISES, PAGES 95–98GUIDED PRACTICE, PAGE 95
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
Copyright © by Holt, Rinehart and Winston. 34 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 343660 a107_Chapter 2_021-058.indd 34 3/14/07 2:21:25 PM3/14/07 2:21:25 PM
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
PRACTICE AND PROBLEM SOLVING, PAGES 95–97
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
Copyright © by Holt, Rinehart and Winston. 35 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 353660 a107_Chapter 2_021-058.indd 35 3/14/07 2:21:27 PM3/14/07 2:21:27 PM
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)
Copyright © by Holt, Rinehart and Winston. 36 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 363660 a107_Chapter 2_021-058.indd 36 3/14/07 2:21:29 PM3/14/07 2:21:29 PM
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
Copyright © by Holt, Rinehart and Winston. 37 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 373660 a107_Chapter 2_021-058.indd 37 3/14/07 2:21:30 PM3/14/07 2:21:30 PM
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
CHALLENGE AND EXTEND, PAGE 98
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
Copyright © by Holt, Rinehart and Winston. 38 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 383660 a107_Chapter 2_021-058.indd 38 3/14/07 2:21:32 PM3/14/07 2:21:32 PM
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.
SPIRAL STANDARDS REVIEW, PAGE 98
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.
Copyright © by Holt, Rinehart and Winston. 39 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 393660 a107_Chapter 2_021-058.indd 39 3/14/07 2:21:34 PM3/14/07 2:21:34 PM
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
Copyright © by Holt, Rinehart and Winston. 40 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 403660 a107_Chapter 2_021-058.indd 40 3/14/07 2:21:35 PM3/14/07 2:21:35 PM
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
CHECK IT OUT! PAGES 102–104
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.
THINK AND DISCUSS, PAGE 104
1. Multiply both sides by 4; use cross products.
2.
EXERCISES, PAGES 105–107GUIDED PRACTICE, PAGE 105
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
Copyright © by Holt, Rinehart and Winston. 41 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 413660 a107_Chapter 2_021-058.indd 41 3/14/07 2:21:37 PM3/14/07 2:21:37 PM
PRACTICE AND PROBLEM SOLVING, PAGES 105–107
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%
Copyright © by Holt, Rinehart and Winston. 42 Holt Algebra 1All rights reserved.
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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
CHALLENGE AND EXTEND, PAGE 107
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
SPIRAL STANDARDS REVIEW, PAGE 107
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
CHECK IT OUT! PAGES 109–110
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
Copyright © by Holt, Rinehart and Winston. 43 Holt Algebra 1All rights reserved.
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THINK AND DISCUSS, PAGE 111
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.
EXERCISES, PAGES 111–113GUIDED PRACTICE, PAGE 111
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
PRACTICE AND PROBLEM SOLVING, PAGES 111–112
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
Copyright © by Holt, Rinehart and Winston. 44 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 443660 a107_Chapter 2_021-058.indd 44 3/14/07 2:21:41 PM3/14/07 2:21:41 PM
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.
CHALLENGE AND EXTEND, PAGE 113
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
Copyright © by Holt, Rinehart and Winston. 45 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 453660 a107_Chapter 2_021-058.indd 45 3/14/07 2:21:44 PM3/14/07 2:21:44 PM
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
SPIRAL STANDARDS REVIEW, PAGE 113
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
CHECK IT OUT! PAGES 115–116
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.
THINK AND DISCUSS, PAGE 116
1. Multiply both sides by 5; split into two cases and solve each one.
2.
Copyright © by Holt, Rinehart and Winston. 46 Holt Algebra 1All rights reserved.
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EXERCISES, PAGES 117–119GUIDED PRACTICE, PAGE 117
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 ✓
Copyright © by Holt, Rinehart and Winston. 47 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 473660 a107_Chapter 2_021-058.indd 47 3/14/07 2:21:48 PM3/14/07 2:21:48 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 ✓
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.
PRACTICE AND PROBLEM SOLVING, PAGES 117–119
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 ✓
Copyright © by Holt, Rinehart and Winston. 48 Holt Algebra 1All rights reserved.
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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.
Copyright © by Holt, Rinehart and Winston. 49 Holt Algebra 1All rights reserved.
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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
Copyright © by Holt, Rinehart and Winston. 50 Holt Algebra 1All rights reserved.
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CHALLENGE AND EXTEND, PAGE 119
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 ✓
SPIRAL STANDARDS REVIEW, PAGE 119
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
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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.
Copyright © by Holt, Rinehart and Winston. 52 Holt Algebra 1All rights reserved.
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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.
LESSON 2-2, PAGE 123
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
Copyright © by Holt, Rinehart and Winston. 53 Holt Algebra 1All rights reserved.
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LESSON 2-3, PAGE 123
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.
LESSON 2-4, PAGE 124
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
Copyright © by Holt, Rinehart and Winston. 54 Holt Algebra 1All rights reserved.
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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%
LESSON 2-6, PAGE 125
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
LESSON 2-7, PAGE 125
74. Case 1x + 6 = 21 _____ - 6 ___ - 6 x = 15
Case 2x + 6 = -21 _____ - 6 ____ - 6 x = -27
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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
Copyright © by Holt, Rinehart and Winston. 56 Holt Algebra 1All rights reserved.
3660 a107_Chapter 2_021-058.indd 563660 a107_Chapter 2_021-058.indd 56 3/14/07 2:22:03 PM3/14/07 2:22:03 PM
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
Copyright © by Holt, Rinehart and Winston. 57 Holt Algebra 1All rights reserved.
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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
Copyright © by Holt, Rinehart and Winston. 58 Holt Algebra 1All rights reserved.
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