2.8E: Exercises for Section 2.8
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In Exercises 1 - 8, use both interval and set-builder notation to describe the intersection of the two intervals shown on the graph. Also, sketch the graph of the intersection on the real number line.
Exercise 1
- Answer
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The intersection is the set of points that are in both intervals (shaded on both graphs). Graph of the intersection:
Set-Builder Notation: {x|x≥1}
Interval Notation: [1,∞)
Exercise 2
Exercise 3
- Answer
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There are no points that are in both intervals (shaded in both), so there is no intersection. Graph of the intersection:
no intersection
Set-Builder Notation: {}
Interval Notation: none
Exercise 4
Exercise 5
- Answer
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The intersection is the set of points that are in both intervals (shaded in both). Graph of the intersection:
Set-Builder Notation: {x|−6≤x≤2}
Interval Notation: [−6,2]
Exercise 6
Exercise 7
- Answer
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The intersection is the set of points that are in both intervals (shaded in both). Graph of the intersection:
Set-Builder Notation: {x|x≥9}
Interval Notation: [9,∞)
Exercise 8
In Exercises 9 - 16, use both interval notation and set-builder notation to describe the union of the two intervals shown on the graph. Also, sketch the graph of the union on the real number line.
Exercise 9
- Answer
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The union is the set of all points that are in one interval or the other (shaded in either graph). Graph of the union:
Set-Builder Notation: {x|x≤−8}
Interval Notation: (−∞,−8]
Exercise 10
Exercise 11
- Answer
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The union is the set of all points that are in one interval or the other (shaded in either graph). Graph of the union:
Set-Builder Notation: {x|x≤9 or x>15}
Interval Notation: (−∞,9]∪(15,∞)
Exercise 12
Exercise 13
- Answer
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The union is the set of all points that are in one interval or the other (shaded in either). Graph of the union:
Set-Builder Notation: {x|x<3}
Interval Notation: (−∞,3)
Exercise 14
Exercise 15
- Answer
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The union is the set of all points that are in one interval or the other (shaded in either). Graph of the union:
Set-Builder Notation: {x|x≥9}
Interval Notation: [9,∞)
Exercise 16
In Exercises 17 - 32, use interval notation to describe the given set. Also, sketch the graph of the set on the real number line.
Exercise 17
{x|x≥−6 and x>−5}
- Answer
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This set is the same as {x|x>−5}, which is (−5,∞) in interval notation. Graph of the set:
Exercise 18
{x|x≤6 and x≥4}
Exercise 19
{x|x≥−1 or x<3}
- Answer
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Every real number is in one or the other of the two intervals. Therefore, the set is the set of all real numbers (−∞,∞). Graph of the set:
Exercise 20
{x|x>−7 and x>−4}
Exercise 21
{x|x≥−1 or x>6}
- Answer
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This set is the same as {x|x≥−1}, which is [−1,∞) in interval notation. Graph of the set:
Exercise 22
{x|x≥7 or x<−2}
Exercise 23
{x|x≥6 or x>−3}
- Answer
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This set is the same as {x|x>−3}, which is (−3,∞) in interval notation. Graph of the set:
Exercise 24
{x|x≤1 or x>0}
Exercise 25
{x|x<2 and x<−7}
- Answer
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This set is the same as {x|x<−7}, which is (−∞,−7) in interval notation. Graph of the set:
Exercise 26
{x|x≤−3 and x<−5}
Exercise 27
{x|x≤−3 or x≥4}
- Answer
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This set is the union of two intervals, (−∞,−3]∪[4,∞). Graph of the set:
Exercise 28
{x|x<11 or x≤8}
Exercise 29
{x|x≥5 and x≤1}
- Answer
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There are no numbers that satisfy both inequalities. Thus, there is no intersection. Graph of the set:
Exercise 30
{x|x<5 or x<10}
Exercise 31
{x|x≤5 and x≥−1}
- Answer
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This set is the same as {x|−1≤x≤5}, which is [−1, 5] in interval notation. Graph of the set
Exercise 32
{x|x>−3 and x<−6}
In Exercises 33 - 44, solve the inequality. Express your answer in both interval and set-builder notations, and graph the solution on a number line. Answers include worked out solutions.
Exercise 33
−8x−3≤−16x−1
- Answer
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−8x−3≤−16x−1⟹−8x+16x≤−1+3⟹8x≤2⟹x≤14
Thus, the solution interval is
Set-Builder Notation: {x|x≤14}
Interval Notation: (−∞,14]
Exercise 34
6x−6>3x+3
Exercise 35
−12x+5≤−3x−4
- Answer
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−12x+5≤−3x−4⟹−12x+3x≤−4−5⟹−9x≤−9⟹x≥1
Thus, the solution interval is
Set-Builder Notation: \(\\{x \, | \, x\geq 1\}\)
Interval Notation: [1,∞)
Exercise 36
7x+3≤−2x−8
Exercise 37
−11x−9<−3x+1
- Answer
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−11x−9<−3x+1⟹−11x+3x<1+9⟹−8x<10⟹x>−54
Thus, the solution interval is
Set-Builder Notation: {x|x>−54}
Interval Notation: (−54,∞)
Exercise 38
4x−8≥−4x−5
Exercise 39
4x−5>5x−7
- Answer
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4x−5>5x−7⟹4x−5x>−7+5⟹−x>−2⟹x<2
Thus, the solution interval isSet-Builder Notation: {x|x<2}
Interval Notation: (−∞,2)
Exercise 40
−14x+4>−6x+8
Exercise 41
2x−1>7x+2
- Answer
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2x−1>7x+2⟹2x−7x>2+1⟹−5x>3⟹x<−35
Thus, the solution interval isSet-Builder Notation: {x|x<−35}
Interval Notation: (−∞,−35)
Exercise 42
−3x−2>−4x−9
Exercise 43
−3x+3<−11x−3
- Answer
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−3x+3<−11x−3⟹−3x+11x<−3−3⟹8x<−6⟹x<−34
Thus, the solution interval isSet-Builder Notation: {x|x<−34}
Interval Notation: (−∞,−34)
Exercise 44
6x+3<8x+8
In Exercises 45-53, solve the compound inequality. Express your answer in both interval and set-builder notations, and graph the solution on a number line. Worked out solutions are provided in the Answers for #45 and #47. Only the interval notation version of the answer is given for Answers in #49-82.
Exercise 45
2x−1<4 or 7x+1≥−4
- Answer
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2x−1<4 or 7x+1≥−4⟹2x<5or7x≥−5⟹x<52orx≥−57
For the union, shade anything shaded in either graph. The solution is the set of all real numbers. (−∞,∞).
Set-Builder Notation: {x|x∈R}
Interval Notation: (−∞,∞)
Exercise 46
−8x+9<−3 and −7x+1>3
Exercise 47
−6x−4<−4 and −3x+7≥−5
- Answer
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−6x−4<−4 and −3x+7≥−5⟹−6x<0and−3x≥−12⟹x>0andx≤4⟹0<x≤4
The intersection is all points shaded in both graphs, so the solution is
Set-Builder Notation: {x|0<x≤4}
Interval Notation: (0,4]
Exercise 48
−3x+3≤8 and −3x−6>−6
Exercise 49
8x+5≤−1 and 4x−2>−1
- Answer
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No Solution
Set-Builder Notation: {}
Exercise 50
−x−1<7 and −6x−9≥8
Exercise 51
−3x+8≤−5 or −2x−4≥−3
- Answer
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No Solution
Set-Builder Notation: {}
Exercise 52
−6x−7<−3 and −8x≥3
Exercise 53
9x−9≤9 and 5x>−1
- Answer
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Interval Notation: (−15,2]
Exercise 2.8E.54
−7x+3<−3 or −8x≥2
Exercise 2.8E.55
3x−5<4 and −x+9>3
- Answer
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Interval Notation: (−∞,3)
Exercise 2.8E.56
−8x−6<5 or 4x−1≥3
Exercise 2.8E.57
9x+3≤−5 or −2x−4≥9
- Answer
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Interval Notation: (−∞,−89)
Exercise 2.8E.58
−7x+6<−4 or −7x−5>7
Exercise 2.8E.59
4x−2≤2 or 3x−9≥3
- Answer
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Add texts here. (−∞,1]∪[4,∞)
Exercise 2.8E.60
−5x+5<−4 or −5x−5≥−5
Exercise 2.8E.61
5x+1<−6 and 3x+9>−4
- Answer
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Interval Notation: (−133,−75)
Exercise 2.8E.62
7x+2<−5 or 6x−9≥−7
Exercise 2.8E.63
−7x−7<−2 and 3x≥3
- Answer
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Interval Notation: [1,∞)
Exercise 2.8E.64
4x+1<0 or 8x+6>9
Exercise 2.8E.65
7x+8<−3 and 8x+3≥−9
- Answer
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No Solution
Set-Builder Notation: {}
Exercise 2.8E.66
3x<2 and −7x−8≥3
Exercise 2.8E.67
−5x+2≤−2 and −6x+2≥3
- Answer
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No Solution
Set-Builder Notation: {}
Exercise 2.8E.68
4x−1≤8 or 3x−9>0
Exercise 2.8E.69
2x−5≤1 and 4x+7>7
- Answer
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(0,3]
Exercise 2.8E.70
3x+1<0 or 5x+5>−8
Exercise 2.8E.71
−8x+7≤9 or −5x+6>−2
- Answer
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(−∞,∞)
Exercise 2.8E.72
x−6≤−5 and 6x−2>−3
Exercise 2.8E.73
−4x−8<4 or −4x+2>3
- Answer
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(−∞,∞)
Exercise 2.8E.74
9x−5<2 or −8x−5≥−6
Exercise 2.8E.75
−9x−5≤−3 or x+1>3
- Answer
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(−29,∞)
Exercise 2.8E.76
−5x−3≤6 and 2x−1≥6
Exercise 2.8E.77
−1≤−7x−3≤2
- Answer
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Interval Notation: [−57,−27]
Exercise 2.8E.78
0<5x−5<9
Exercise 79
5<9x−3≤6
- Answer
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Graph of the solution:
Set-Builder Notation: {x|89<x≤1}
Interval Notation: (89,1]
Exercise 2.8E.80
−6<7x+3≤2
Exercise 2.8E.81
−2<−7x+6<6
- Answer
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Interval Notation: (0,87)
Exercise 82
−9<−2x+5≤1
In Exercises 83-94, 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. Only the interval notation version of the answer is given for Answers in #83-94.
Exercise 2.8E.83
−13<x2+14<13
- Answer
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Interval Notation: (−76,16)
Exercise 2.8E.84
−15<x2−14<15
Exercise 2.8E.85
−12<13−x2<12
- Answer
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Interval Notation: (−13,53)
Exercise 86
−23≤12−x5≤23
Exercise 2.8E.87
−1<x−x+15<2
- Answer
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Interval Notation: (−1,114)
Exercise 2.8E.88
−2<x−2x−13<4
Exercise 2.8E.89
−2<x+12−x+13≤2
- Answer
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Interval Notation: (−13,11]
Exercise 2.8E.90
−3<x−13−2x−15≤2
Exercise 2.8E.91
x<4−x<5
- Answer
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Interval Notation: (−1,2)
Exercise 2.8E.92
−x<2x+3≤7
Exercise 2.8E.93
−x<x+5≤11
- Answer
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Interval Notation: (−52,6]
Exercise 2.8E.94
−2x<3−x≤8
More Practice with Compound Inequalities
For each graph below, describe the interval (a) using set-builder notation and (b) using interval notation.
1. 2. 3. 4. |
5. 6. 7. 8. |
- Answer
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1. (a) {x|3≤x≤4} (b) [3,4]
3. (a) {x|−4<x<1} (b) (−4,1)5. (a) {x|−1≤x≤5} (b) [−1,5]
7. (a) {x|−1<x≤5} (b) (−1,5]