# 2.8E: Exercises for Section 2.8

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- 57739

**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**-
The intersection is the set of points that are in both intervals (shaded on both graphs). Graph of the intersection:

**Set-Builder Notation:**\(\{x \, | \, x \geq 1\}\)**Interval Notation:**\([1, \infty)\)

Exercise 2

Exercise 3

**Answer**-
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**-
The intersection is the set of points that are in both intervals (shaded in both). Graph of the intersection:

**Set-Builder Notation:**\(\{x \, | \, -6 \leq x \leq 2\}\)**Interval Notation:**\([-6,2]\)

Exercise 6

Exercise 7

**Answer**-
The intersection is the set of points that are in both intervals (shaded in both). Graph of the intersection:

**Set-Builder Notation:**\(\{x \, | \, x \geq 9\}\)**Interval Notation:**\([9, \infty)\)

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**-
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:**\(\{x \, | \, x \leq-8\}\)**Interval Notation:**\((-\infty,-8]\)

Exercise 10

Exercise 11

**Answer**-
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:**\(\{x \, | \, x \leq 9 \text { or } x>15\}\)**Interval Notation:**\((-\infty, 9] \cup(15, \infty)\)

Exercise 12

Exercise 13

**Answer**-
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:**\((-\infty, 3)\)

Exercise 14

Exercise 15

**Answer**-
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:**\(\{x \, | \, x \geq 9\}\)**Interval Notation:**\([9, \infty)\)

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

\(\{x \, | \, x \geq-6 \text { and } x>-5\}\)

**Answer**-
This set is the same as \(\{x \, | \, x>-5\}\), which is \((-5, \infty)\) in interval notation. Graph of the set:

Exercise 18

\(\{x \, | \, x \leq 6 \text { and } x \geq 4\}\)

Exercise 19

\(\{x \, | \, x \geq-1 \text { or } x<3\}\)

**Answer**-
Every real number is in one or the other of the two intervals. Therefore, the set is the set of all real numbers \((-\infty, \infty)\). Graph of the set:

Exercise 20

\(\{x \, | \, x>-7 \text { and } x>-4\}\)

Exercise 21

\(\{x \, | \, x \geq -1 \text { or } x>6\}\)

**Answer**-
This set is the same as \(\{x \, | \, x \geq-1\}\), which is \([-1, \infty)\) in interval notation. Graph of the set:

Exercise 22

\(\{x \, | \, x \geq 7 \text { or } x<-2\}\)

Exercise 23

\(\{x \, | \, x \geq 6 \text { or } x>-3\}\)

**Answer**-
This set is the same as \(\{x \, | \, x>-3\}\), which is \((-3, \infty)\) in interval notation. Graph of the set:

Exercise 24

\(\{x \, | \, x \leq 1 \text { or } x>0\}\)

Exercise 25

\(\{x \, | \, x<2 \text { and } x<-7\}\)

**Answer**-
This set is the same as \(\{x \, | \, x<-7\}\), which is \((-\infty,-7)\) in interval notation. Graph of the set:

Exercise 26

\(\{x \, | \, x \leq-3 \text { and } x<-5\}\)

Exercise 27

\(\{x \, | \, x \leq-3 \text { or } x \geq 4\}\)

**Answer**-
This set is the union of two intervals, \((-\infty,-3] \cup[4, \infty)\). Graph of the set:

Exercise 28

\(\{x \, | \, x<11 \text { or } x \leq 8\}\)

Exercise 29

\(\{x \, | \, x \geq 5 \text { and } x \leq 1\}\)

**Answer**-
There are no numbers that satisfy both inequalities. Thus, there is no intersection. Graph of the set:

Exercise 30

\(\{x \, | \, x<5 \text { or } x<10\}\)

Exercise 31

\(\{x \, | \, x \leq 5 \text { and } x \geq-1\}\)

**Answer**-
This set is the same as \(\{x \, | \, -1 \leq x \leq 5\}\), which is [−1, 5] in interval notation. Graph of the set

Exercise 32

\(\{x \, | \, x>-3 \text { and } x<-6\}\)

**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.**

Exercise 33

\(-8 x-3 \leq-16 x-1\)

**Answer**-
\[\begin{aligned} & -8 x-3 \leq-16 x-1 \\ \Longrightarrow \quad & − 8x + 16x \leq −1 + 3 \\ \Longrightarrow \quad& 8x \leq 2 \\ \Longrightarrow \quad & x \leq \frac{1}{4}\end{aligned}\]

Thus, the solution interval is

**Set-Builder Notation:**\(\{x \, | \, x \leq \frac{1}{4}\}\)**Interval Notation:**\((−\infty, \frac{1}{4}]\)

Exercise 34

\(6 x-6>3 x+3\)

Exercise 35

\(-12 x+5 \leq-3 x-4\)

**Answer**-
\[\begin{aligned} & -12 x+5 \leq-3 x-4 \\ \Longrightarrow \quad & -12x + 3x \leq −4 − 5 \\ \Longrightarrow \quad& -9x \leq -9 \\ \Longrightarrow \quad & x \geq 1\end{aligned}\]

Thus, the solution interval is

**Set-Builder Notation:**\(\\{x \, | \, x\geq 1\}\)**Interval Notation:**\([1,\infty)\)

Exercise 36

\(7 x+3 \leq-2 x-8\)

Exercise 37

\(-11 x-9<-3 x+1\)

**Answer**-
\[\begin{aligned} & − 11x − 9 < −3x + 1 \\ \Longrightarrow \quad & − 11x + 3x < 1 + 9 \\ \Longrightarrow \quad& − 8x < 10 \\ \Longrightarrow \quad & x > -\frac{5}{4}\end{aligned}\]

Thus, the solution interval is

**Set-Builder Notation:**\(\{x \, | \, x >−\frac{5}{4} \}\)**Interval Notation:**\((−\frac{5}{4} ,\infty) \)

Exercise 38

\(4 x-8 \geq-4 x-5\)

Exercise 39

\(4 x-5>5 x-7\)

**Answer**-
\[\begin{align*} & 4x − 5 > 5x − 7\\ \Longrightarrow \quad & 4x − 5x > −7 + 5 \\ \Longrightarrow \quad& − x > −2 \\ \Longrightarrow \quad &x < 2\end{align*}\]

Thus, the solution interval is**Set-Builder Notation:**\(\{x \, | \, x < 2\}\)**Interval Notation:**\((−\infty, 2) \)

Exercise 40

\(-14 x+4>-6 x+8\)

Exercise 41

\(2 x-1>7 x+2\)

**Answer**-
\[\begin{aligned} & 2x − 1 > 7x + 2\\ \Longrightarrow \quad & 2x − 7x > 2 + 1 \\ \Longrightarrow \quad& − 5x > 3 \\ \Longrightarrow \quad &x < −\frac{3}{5}\end{aligned}\]

Thus, the solution interval is**Set-Builder Notation:**\(\{x \, | \, x < −\frac{3}{5}\}\)**Interval Notation:**\((−\infty, −\frac{3}{5})\)

Exercise 42

\(-3 x-2>-4 x-9\)

Exercise 43

\(-3 x+3<-11 x-3\)

**Answer**-
\[\begin{aligned} & − 3x + 3 < −11x − 3\\ \Longrightarrow \quad & − 3x + 11x < −3 − 3 \\ \Longrightarrow \quad& 8x < −6 \\ \Longrightarrow \quad &x < -\frac{3}{4}\end{aligned}\]

Thus, the solution interval is**Set-Builder Notation:**\(\{x \, | \, x < −\frac{3}{4}\}\)**Interval Notation:**\((−\infty, −\frac{3}{4})\)

Exercise 44

\(6 x+3<8 x+8\)

**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.**

Exercise 45

\(2 x-1<4\) or \(7 x+1 \geq-4\)

**Answer**-
\[\begin{aligned} & 2x − 1 < 4 \text{ or } 7x + 1 \geq −4\\ \Longrightarrow \quad & 2x < 5\quad \text{or}\quad 7x \geq −5 \\ \Longrightarrow \quad&x<\frac{5}{2}\quad\text{or}\quad x\geq-\frac{5}{7}\end{aligned}\]

For the union, shade anything shaded in either graph. The solution is the set of all real numbers. \((−\infty,\infty)\).

**Set-Builder Notation:**\(\{x \, | \, x \in \mathbb{R}\}\)**Interval Notation:**\((−\infty, \infty)\)

Exercise 46

\(-8 x+9<-3\) and \(-7 x+1>3\)

Exercise 47

\(-6 x-4<-4\) and \(-3 x+7 \geq-5\)

**Answer**-
\[\begin{aligned} & − 6x − 4 < −4 \text{ and } − 3x + 7 \geq −5\\ \Longrightarrow \quad & -6x < 0\quad \text{and}\quad -3x \geq −12 \\ \Longrightarrow \quad&x>0\quad\text{and}\quad x\leq4 \\ \Longrightarrow \quad & 0< x \leq 4 \end{aligned}\]

The intersection is all points shaded in both graphs, so the solution is

**Set-Builder Notation:**\(\{x \, | \, 0 < x \leq 4\}\)**Interval Notation:**\((0, 4]\)

Exercise 48

\(-3 x+3 \leq 8\) and \(-3 x-6>-6\)

Exercise 49

\(8 x+5 \leq-1\) and \(4 x-2>-1\)

**Answer**-
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Exercise 50

\(-x-1<7\) and \(-6 x-9 \geq 8\)

Exercise 51

\(-3 x+8 \leq-5\) or \(-2 x-4 \geq-3\)

**Answer**-
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Exercise 52

\(-6 x-7<-3\) and \(-8 x \geq 3\)

Exercise 53

\(9 x-9 \leq 9\) and \(5 x>-1\)

**Answer**-
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Exercise \(\PageIndex{54}\)

\(-7 x+3<-3\) or \(-8 x \geq 2\)

Exercise \(\PageIndex{55}\)

\(3 x-5<4\) and \(-x+9>3\)

**Answer**-
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Exercise \(\PageIndex{56}\)

\(-8 x-6<5\) or \(4 x-1 \geq 3\)

Exercise \(\PageIndex{57}\)

\(9 x+3 \leq-5\) or \(-2 x-4 \geq 9\)

**Answer**-
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Exercise \(\PageIndex{58}\)

\(-7 x+6<-4\) or \(-7 x-5>7\)

Exercise \(\PageIndex{59}\)

\(4 x-2 \leq 2\) or \(3 x-9 \geq 3\)

**Answer**-
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Exercise \(\PageIndex{60}\)

\(-5 x+5<-4\) or \(-5 x-5 \geq-5\)

Exercise \(\PageIndex{61}\)

\(5 x+1<-6\) and \(3 x+9>-4\)

**Answer**-
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Exercise \(\PageIndex{62}\)

\(7 x+2<-5\) or \(6 x-9 \geq-7\)

Exercise \(\PageIndex{63}\)

\(-7 x-7<-2\) and \(3 x \geq 3\)

**Answer**-
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Exercise \(\PageIndex{64}\)

\(4 x+1<0\) or \(8 x+6>9\)

Exercise \(\PageIndex{65}\)

\(7 x+8<-3\) and \(8 x+3 \geq-9\)

**Answer**-
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Exercise \(\PageIndex{66}\)

\(3 x<2\) and \(-7 x-8 \geq 3\)

Exercise \(\PageIndex{67}\)

\(-5 x+2 \leq-2\) and \(-6 x+2 \geq 3\)

**Answer**-
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Exercise \(\PageIndex{68}\)

\(4 x-1 \leq 8\) or \(3 x-9>0\)

Exercise \(\PageIndex{69}\)

\(2 x-5 \leq 1\) and \(4 x+7>7\)

**Answer**-
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Exercise \(\PageIndex{70}\)

\(3 x+1<0\) or \(5 x+5>-8\)

Exercise \(\PageIndex{71}\)

\(-8 x+7 \leq 9\) or \(-5 x+6>-2\)

**Answer**-
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Exercise \(\PageIndex{72}\)

\(x-6 \leq-5\) and \(6 x-2>-3\)

Exercise \(\PageIndex{73}\)

\(-4 x-8<4\) or \(-4 x+2>3\)

**Answer**-
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Exercise \(\PageIndex{74}\)

\(9 x-5<2\) or \(-8 x-5 \geq-6\)

Exercise \(\PageIndex{75}\)

\(-9 x-5 \leq-3\) or \(x+1>3\)

**Answer**-
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Exercise \(\PageIndex{76}\)

\(-5 x-3 \leq 6\) and \(2 x-1 \geq 6\)

Exercise \(\PageIndex{77}\)

\(-1 \leq-7 x-3 \leq 2\)

**Answer**-
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Exercise \(\PageIndex{78}\)

\(0<5 x-5<9\)

Exercise 79

\(5<9 x-3 \leq 6\)

**Answer**-
Graph of the solution:

**Set-Builder Notation:**\(\{x \, | \, \frac{8}{9} \lt x \leq 1\}\)**Interval Notation:**\( (\frac{8}{9},1] \)

Exercise \(\PageIndex{80}\)

\(-6<7 x+3 \leq 2\)

Exercise \(\PageIndex{81}\)

\(-2<-7 x+6<6\)

**Answer**-
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Exercise 82

\(-9<-2 x+5 \leq 1\)

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.

Exercise \(\PageIndex{83}\)

\(-\frac{1}{3}<\frac{x}{2}+\frac{1}{4}<\frac{1}{3}\)

**Answer**-
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Exercise \(\PageIndex{84}\)

\(-\frac{1}{5}<\frac{x}{2}-\frac{1}{4}<\frac{1}{5}\)

Exercise \(\PageIndex{85}\)

\(-\frac{1}{2}<\frac{1}{3}-\frac{x}{2}<\frac{1}{2}\)

**Answer**-
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Exercise 86

\(-\frac{2}{3} \leq \frac{1}{2}-\frac{x}{5} \leq \frac{2}{3}\)

Exercise \(\PageIndex{87}\)

\(-1<x-\frac{x+1}{5}<2\)

**Answer**-
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Exercise \(\PageIndex{88}\)

\(-2<x-\frac{2 x-1}{3}<4\)

Exercise \(\PageIndex{89}\)

\(-2<\frac{x+1}{2}-\frac{x+1}{3} \leq 2\)

**Answer**-
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Exercise \(\PageIndex{90}\)

\(-3<\frac{x-1}{3}-\frac{2 x-1}{5} \leq 2\)

Exercise \(\PageIndex{91}\)

\(x<4-x<5\)

**Answer**-
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Exercise \(\PageIndex{92}\)

\(-x<2 x+3 \leq 7\)

Exercise \(\PageIndex{93}\)

\(-x<x+5 \leq 11\)

**Answer**-
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Exercise \(\PageIndex{94}\)

\(−2x < 3 − x \leq 8\)