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Section 1.4
Compound Inequalities
63
1.4 Exercises
In Exercises 1-12, solve the inequality.
Express your answer in both interval and
set notations, and shade the solution on
a number line.
1.
2.
8x + 5 ≤ −1 and 4x − 2 > −1
18.
−x − 1 < 7 and − 6x − 9 ≥ 8
19.
−3x + 8 ≤ −5 or − 2x − 4 ≥ −3
20.
−6x − 7 < −3 and − 8x ≥ 3
21.
9x − 9 ≤ 9 and 5x > −1
22.
−7x + 3 < −3 or − 8x ≥ 2
23.
3x − 5 < 4 and − x + 9 > 3
24.
−8x − 6 < 5 or 4x − 1 ≥ 3
25.
9x + 3 ≤ −5 or − 2x − 4 ≥ 9
26.
−7x + 6 < −4 or − 7x − 5 > 7
27.
4x − 2 ≤ 2 or 3x − 9 ≥ 3
28.
−5x + 5 < −4 or − 5x − 5 ≥ −5
29.
5x + 1 < −6 and 3x + 9 > −4
30.
7x + 2 < −5 or 6x − 9 ≥ −7
31.
−7x − 7 < −2 and 3x ≥ 3
32.
4x + 1 < 0 or 8x + 6 > 9
33.
7x + 8 < −3 and 8x + 3 ≥ −9
34.
3x < 2 and − 7x − 8 ≥ 3
35.
−5x + 2 ≤ −2 and − 6x + 2 ≥ 3
36.
4x − 1 ≤ 8 or 3x − 9 > 0
37.
2x − 5 ≤ 1 and 4x + 7 > 7
38.
3x + 1 < 0 or 5x + 5 > −8
−8x − 3 ≤ −16x − 1
6x − 6 > 3x + 3
3.
−12x + 5 ≤ −3x − 4
4.
7x + 3 ≤ −2x − 8
5.
−11x − 9 < −3x + 1
6.
4x − 8 ≥ −4x − 5
7.
4x − 5 > 5x − 7
8.
−14x + 4 > −6x + 8
9.
2x − 1 > 7x + 2
10.
−3x − 2 > −4x − 9
11.
−3x + 3 < −11x − 3
12.
6x + 3 < 8x + 8
In Exercises 13-50, solve the compound
inequality. Express your answer in both
interval and set notations, and shade the
solution on a number line.
1
17.
13.
2x − 1 < 4 or 7x + 1 ≥ −4
14.
−8x + 9 < −3 and − 7x + 1 > 3
15.
−6x − 4 < −4 and − 3x + 7 ≥ −5
16.
−3x + 3 ≤ 8 and − 3x − 6 > −6
Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/
Version: Fall 2007
64
Chapter 1
Preliminaries
39.
−8x + 7 ≤ 9 or − 5x + 6 > −2
40.
x − 6 ≤ −5 and 6x − 2 > −3
41.
−4x − 8 < 4 or − 4x + 2 > 3
42.
9x − 5 < 2 or − 8x − 5 ≥ −6
43.
−9x − 5 ≤ −3 or x + 1 > 3
44.
−5x − 3 ≤ 6 and 2x − 1 ≥ 6
45.
−1 ≤ −7x − 3 ≤ 2
46.
0 < 5x − 5 < 9
47.
5 < 9x − 3 ≤ 6
48.
−6 < 7x + 3 ≤ 2
49.
−2 < −7x + 6 < 6
50.
−9 < −2x + 5 ≤ 1
In Exercises 51-62, solve the given inequality for x. Graph the solution set on
a number line, then use interval and setbuilder notation to describe the solution
set.
51.
1
x 1
1
− < + <
3
2 4
3
52.
x 1
1
1
− < − <
5
2 4
5
53.
1
1 x
1
− < − <
2
3 2
2
54.
2
1 x
2
− ≤ − ≤
3
2 5
3
55.
−1 < x −
x+1
<2
5
56.
−2 < x −
2x − 1
<4
3
57.
−2 <
x+1 x+1
−
≤2
2
3
Version: Fall 2007
x − 1 2x − 1
−
≤2
3
5
58.
−3 <
59.
x<4−x<5
60.
−x < 2x + 3 ≤ 7
61.
−x < x + 5 ≤ 11
62.
−2x < 3 − x ≤ 8
63. Aeron has arranged for a demonstration of “How to make a Comet” by
Professor O’Commel. The wise professor has asked Aeron to make sure the
auditorium stays between 15 and 20 degrees Celsius (C). Aeron knows the thermostat is in Fahrenheit (F) and he also
knows that the conversion formula between the two temperature scales is C =
(5/9)(F − 32).
a) Setting up the compound inequality
for the requested temperature range
in Celsius, we get 15 ≤ C ≤ 20. Using the conversion formula above, set
up the corresponding compound inequality in Fahrenheit.
b) Solve the compound inequality in part
(a) for F. Write your answer in set
notation.
c) What are the possible temperatures
(integers only) that Aeron can set the
thermostat to in Fahrenheit?
Section 1.4
Compound Inequalities
65
1.4 Answers
1.
(−∞, 14 ] = {x|x ≤ 14 }
S 13
19. −∞, − 21
3 ,∞
={x|x ≤ − 12 or x ≥ 13
3 }
1
4
3.
− 12
13
3
[1, ∞) = {x|x ≥ 1}
21.
(− 15 , 2] = {x| −
1
5
< x ≤ 2}
1
5.
(− 45 , ∞)
= {x|x >
23.
− 54
2
− 15
− 54 }
(−∞, 3) = {x|x < 3}
3
7.
(−∞, 2) = {x|x < 2}
25.
2
9.
(−∞, − 35 )
= {x|x <
27.
4}
(−∞, − 34 ) = {x|x < − 34 }
− 34
13.
− 89
− 35 }
− 35
11.
(−∞, 1]
S
[4, ∞) = {x|x ≤ 1 or x ≥
1
29.
4
7
(− 13
3 , − 5 ) = {x| −
− 13
3
(−∞, ∞) = {all real numbers}
31.
15.
(−∞, − 98 ] = {x|x ≤ − 89 }
13
3
< x < − 75 }
− 75
[1, ∞) = {x|x ≥ 1}
(0, 4] = {x|0 < x ≤ 4}
1
0
4
33.
17.
no solution
no solution
Version: Fall 2007
66
Chapter 1
Preliminaries
53. (−1/3, 5/3) = {x : −1/3 < x <
5/3}
35.
no solution
37.
(0, 3] = {x|0 < x ≤ 3}
0
39.
3
−1/3
55.
57.
41.
(−∞, ∞) = {all real numbers}
[− 29 , ∞) = {x|x ≥ − 29 }
− 29
45.
[− 75 , − 72 ] = {x| −
− 75
47.
≤ x ≤ − 72 }
− 72
( 89 , 1] = {x| 89 < x ≤ 1}
(−13, 11] = {x : −13 < x ≤ 11}
11
(−1, 2) = {x : −1 < x < 2}
−1
61.
5
7
11/4
−13
59.
43.
(−1, 11/4) = {x : −1 < x < 11/4}
−1
(−∞, ∞) = {all real numbers}
5/3
2
(−5/2, 6] = {x : −5/2 < x ≤ 6}
−5/2
6
63.
a) 15 ≤ 59 (F − 32) ≤ 20
b) {F : 59 ≤ F ≤ 68}
8
9
49.
1
(0, 78 ) = {x|0 < x < 78 }
0
8
7
51. (−7/6, 1/6) = {x : −7/6 < x <
1/6}
−7/6
Version: Fall 2007
1/6
c) {59, 60, 61, 62, 63, 64, 65, 66, 67, 68}