CHAPTER 1 Section 1.1 Solutions ----------------------------------------------------------------------------1.
5x = 35 1 1 ⋅ 5x = ⋅ 35 5 5
2.
4 = −5 + y 5 + 4 = 5 + −5 + y
5.
9=y
4p +5 = 9 4p = 4
5t + 11 = 18 5t = 7 t=7 5
n = 15
6.
−50 = −5t 1 1 − ⋅ ( −50) = − ⋅ ( −5t ) 5 5 10 = t
9.
3x − 5 = 7 3x = 12 x=4
18 = p
11.
p =1
13.
−3 + n = 12 3 + −3 + n = 3 + 12
1 6= p 3 1 3⋅6 = 3⋅ p 3
1 n=3 5 1 5 ⋅ n = 5 ⋅3 5 n = 15 10.
24 = −3x 1 1 − ⋅ 24 = − ⋅ ( −3x ) 3 3 −8 = x
8.
7.
3.
1 1 ⋅ 4t = ⋅ 32 4 4 t =8
x=7
4.
4t = 32
14.
9m − 7 = 11
12.
2x + 4 = 5
9m = 18
2x = 1
m=2
x =1 2
7x + 4 = 21 + 24x 7x = 17 + 24x −17 x = 17 x = −1
59
15.
3x − 5 = 25 + 6 x 3x = 30 + 6 x −3x = 30 x = −10
Chapter 1 16.
17.
5x + 10 = 25 + 2 x 5x = 15 + 2 x 3x = 15 x=5
19.
21.
23.
25.
20 n − 30 = 20 − 5n 20 n = 50 − 5n 25n = 50
18.
14c + 15 = 43 + 7c 14c = 28 + 7c 7c = 28
n=2
20.
4( x − 3) = 2( x + 6) 4 x − 12 = 2 x + 12 2 x = 24 x = 12
c=4
5(2 y − 1) = 2(4 y − 3) 10 y − 5 = 8 y − 6 2 y = −1
y = − 12 22.
−3(4t − 5) = 5(6 − 2t ) −12t + 15 = 30 − 10t −15 = 2t − 15 2 = t
2(3n + 4) = −( n + 2) 6n + 8 = −n − 2 7n = −10 n = − 10 7
24.
2 ( x − 1) + 3 = x − 3 ( x + 1)
4 ( y + 6) − 8 = 2y − 4 ( y + 2)
2 x − 2 + 3 = x − 3x − 3
4 y + 24 − 8 = 2 y − 4 y − 8
2 x + 1 = −2 x − 3 4 x = −4
4 y + 16 = −2 y − 8
x = −1
y = −4
6 y = −24
26.
5 p + 6 ( p + 7) = 3( p + 2) 5 p + 6 p + 42 = 3 p + 6
3z + 15 − 5 = 4 z + 7 z − 14
11p + 42 = 3 p + 6
3z + 10 = 11z − 14 −8z = −24
8 p = −36 p=−
3 ( z + 5) − 5 = 4z + 7 ( z − 2 )
z =3
9 2
60
Section 1.1
27.
28. 7x − (2 x + 3) = x − 2 7x − 2x − 3 = x − 2
3x − (4 x + 2) = x − 5 3x − 4 x − 2 = x − 5
5x − 3 = x − 2
−x − 2 = x − 5
4x = 1 x = 14
3 = 2x 3 = x 2
29.
30. 2 − (4 x + 1) = 3 − (2 x − 1)
5 − (2 x − 3) = 7 − (3x + 5)
2 − 4x − 1 = 3 − 2x + 1
5 − 2 x + 3 = 7 − 3x − 5
1 − 4x = 4 − 2x
8 − 2 x = 2 − 3x x = −6
−3 = 2 x − 32 = x
31.
2a − 9 ( a + 6 ) = 6 ( a + 3 ) − 4 a −7a − 54 = 6a + 18 − 4a −7a − 54 = 2a + 18 −9a = 72 a = −8
32.
25 − 2 + 5 y − 3 ( y + 2 ) = −3 ( 2 y − 5 ) − 5 ( y − 1) − 3 y + 3 25 − [ 2 + 5 y − 3 y − 6 ] = −6 y + 15 − [5 y − 5 − 3 y + 3] 25 − 2 − 5 y + 3 y + 6 = −6 y + 15 − 5 y + 5 + 3 y − 3 29 − 2 y = −8 y + 17 6 y = −12 y = −2
61
Chapter 1 33.
32 − 4 + 6x − 5 ( x + 4 ) = 4 ( 3x + 4 ) − 6 ( 3x − 4 ) + 7 − 4 x 32 − [ 4 + 6x − 5x − 20 ] = 12x + 16 − [18x − 24 + 7 − 4x ] 32 − 4 − 6x + 5x + 20 = 12x + 16 − 18x + 24 − 7 + 4x 48 − x = −2x + 33 x = −15
34.
12 − 3 + 4 m − 6 ( 3m − 2 ) = −7 ( 2 m − 8 ) − 3 ( m − 2 ) + 3m − 5 12 − [3 + 4 m − 18m + 12 ] = −14 m + 56 − 3 [ m − 2 + 3m − 5] 12 − 3 − 4 m + 18m − 12 = −14 m + 56 − 3m + 6 − 9 m + 15 −3 + 14 m = −26 m + 77 40 m = 80 m=2
35.
20 − 4 c − 3 − 6 ( 2c + 3 ) = 5 ( 3c − 2 ) − 2 ( 7c − 8 ) − 4c + 7 20 − 4 [c − 3 − 12c − 18] = 15c − 10 − [14c − 16 − 4c + 7 ] 20 − 4c + 12 + 48c + 72 = 15c − 10 − 14c + 16 + 4c − 7 44c + 104 = 5c − 1 39c = −105 c=
36.
−35 13
46 − 7 − 8 y + 9 ( 6 y − 2 ) = −7 ( 4 y − 7 ) − 2 6 ( 2 y − 3 ) − 4 + 6 y 46 − [7 − 8 y + 54 y − 18] = −28 y + 49 − 2 [12 y − 18 − 4 + 6 y ] 46 − 7 + 8 y − 54 y + 18 = −28 y + 49 − 24 y + 36 + 8 − 12 y −46 y + 57 = −64 y + 93 18 y = 36 y=2
62
Section 1.1
37.
38.
1 1 24 z = 24 z + 3 12 24 2 z = z + 72
1 1 60 m = 60 m + 1 5 60 12 m = m + 60 11m = 60 m=
z = 72
60 11
39.
40.
x 2x 63 = 63 + 4 7 63 9x = 2 x + 252 7 x = 252
a a 22 = 22 + 9 11 22 2a = a + 198 a = 198
x = 36
41.
42.
3x x 5 10 − x = 10 − 5 10 2 6x − 10x = x − 25 −5x = −25
1 1 p 24 p = 24 3 − 24 3 8 p = 72 − p 9 p = 72
x =5
p=8 43.
44.
5y 2y 5 + 84 − 2 y = 84 3 84 7 140 y − 168 y = 2 y + 60 −30 y = 60
5m 3m 4 72 2 m − + = 72 8 72 3 144 m − 45 m = 3m + 96 96 m = 96
y = −2
m =1
63
Chapter 1 45.
46.
c 5 c 4 − 2c = 4 − 4 4 2 c − 8c = 5 − 2c −5c = 5
p 5 8 p + = 8 4 2 8 p + 2 p = 20 10 p = 20
c = −1
p=2 47.
48.
x −3 x − 4 x −6 − = 1− 3 2 6 x −3 x − 4 x −6 6⋅ − = 6 ⋅ 1 − 2 6 3 2( x − 3) − 3( x − 4) = 6 − ( x − 6) 2 x − 6 − 3x + 12 = 6 − x + 6 −x + 6 = −x + 12 6 = 12, which is false. Hence, no solution.
x − 5 x + 2 6x − 1 = − 3 5 15 x −5 x + 2 6x − 1 = 15 ⋅ − 15 ⋅ 1 − 3 15 5 15 − 5( x − 5) = 3( x + 2) − (6 x − 1) 15 − 5x + 25 = 3x + 6 − 6 x + 1 40 − 5x = −3x + 7 33 = 2 x 33 = x 2 1−
49.
50. 4 5 2y − 5 = 2y y 2y 8 − 10 y = 5 −10 y = −3
4 2 3x + 10 = 3x x 3x 12 + 30x = 2 30x = −10
y≠0
x≠0
x = − 13
3 y= 10
51.
52. 1 10 6x 7 − = 6x 6x 3x 42 x − 1 = 20 42 x = 21 x=
5 7 6t = 6t 2 + 3t 6t 7 = 12t + 10 −12t = 3
x≠0
1 2
t=
64
−1 4
t≠0
Section 1.1
53.
54.
2 4 3a − 4 = 3a a 3a 6 − 12a = 4
4 5 x − 2 = x x ≠ 0 x 2x 4 − 2x = 5 2 2x = 4 − 5 2 = 3 2
a≠0
−12a = −2 a=
x = 34
1 6
55.
56.
x 2 + 5 = ( x − 2) ( x − 2) x≠2 x−2 x−2 x + 5( x − 2) = 2 x + 5x − 10 = 2 6x = 12 x=2 No solution since 2 was excluded from the solution set.
n n + 2 = ( n − 5) ( n − 5) n≠5 n −5 n −5 n + 2( n − 5) = n 3n − 10 = n 2 n = 10 n=5 No solution since 5 was excluded from the solution set.
57.
58.
2p 2 ( p − 1) = ( p − 1) 3 + p ≠1 p −1 p −1 2 p = 3( p − 1) + 2 2 p = 3p −3 + 2 2 p = 3 p −1 p =1 No solution since 1 was excluded from the solution set.
8 4t (t + 2) = (t + 2) 3 − t ≠ −2 t+2 t+2 4t = 3(t + 2) − 8 4t = 3t + 6 − 8 t = −2 No solution since −2 was excluded from the solution set.
59.
3x 2 − 4 = ( x + 2) ( x + 2) x ≠ −2 x+2 x+2 3x − 4( x + 2) = 2
−x − 8 = 2 x = −10
65
Chapter 1 99. y1 = 3 ( x + 2 ) − 5x
100. y1 = −5 ( x − 1) − 7
y2 = 3x − 4
y2 = 10 − 9x
x =3
x=2 101. y1 = 2x + 6 y2 = 4 x − 2 x + 8 − 2
102. y1 = 10 − 20 x y2 = 10x − 30 x + 20 − 10
All real numbers All real numbers
72
Section 1.1
103. y1 =
x ( x − 1) x2
104. y2 = 1
y1 =
2x ( x + 3 ) x2
y2 = 2
No solution No solution 105. y1 = 0.035x + 0.029(8706 − x ) y2 = 285.03
106.
x = 5426
x = 7.95
73
y1 =
1 0.45 − 0.75x x
y2 =
1 9