# 2.8E: Exercises for Section 2.8

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

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 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 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 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 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 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 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 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 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$-8 x-3 \leq-16 x-1$$

\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

$$-12 x+5 \leq-3 x-4$$

\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

$$-11 x-9<-3 x+1$$

\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

$$4 x-5>5 x-7$$

\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

$$2 x-1>7 x+2$$

\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

$$-3 x+3<-11 x-3$$

\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 $$7 x+1 \geq-4$$

\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 $$-3 x+7 \geq-5$$

\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 $$4 x-2>-1$$

Add texts here. Do not delete this text first.

Exercise 50

$$-x-1<7$$ and $$-6 x-9 \geq 8$$

Exercise 51

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

Add texts here. Do not delete this text first.

Exercise 52

$$-6 x-7<-3$$ and $$-8 x \geq 3$$

Exercise 53

$$9 x-9 \leq 9$$ and $$5 x>-1$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{54}$$

$$-7 x+3<-3$$ or $$-8 x \geq 2$$

Exercise $$\PageIndex{55}$$

$$3 x-5<4$$ and $$-x+9>3$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{56}$$

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

Exercise $$\PageIndex{57}$$

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

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{58}$$

$$-7 x+6<-4$$ or $$-7 x-5>7$$

Exercise $$\PageIndex{59}$$

$$4 x-2 \leq 2$$ or $$3 x-9 \geq 3$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{60}$$

$$-5 x+5<-4$$ or $$-5 x-5 \geq-5$$

Exercise $$\PageIndex{61}$$

$$5 x+1<-6$$ and $$3 x+9>-4$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{62}$$

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

Exercise $$\PageIndex{63}$$

$$-7 x-7<-2$$ and $$3 x \geq 3$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{64}$$

$$4 x+1<0$$ or $$8 x+6>9$$

Exercise $$\PageIndex{65}$$

$$7 x+8<-3$$ and $$8 x+3 \geq-9$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{66}$$

$$3 x<2$$ and $$-7 x-8 \geq 3$$

Exercise $$\PageIndex{67}$$

$$-5 x+2 \leq-2$$ and $$-6 x+2 \geq 3$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{68}$$

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

Exercise $$\PageIndex{69}$$

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

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{70}$$

$$3 x+1<0$$ or $$5 x+5>-8$$

Exercise $$\PageIndex{71}$$

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

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{72}$$

$$x-6 \leq-5$$ and $$6 x-2>-3$$

Exercise $$\PageIndex{73}$$

$$-4 x-8<4$$ or $$-4 x+2>3$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{74}$$

$$9 x-5<2$$ or $$-8 x-5 \geq-6$$

Exercise $$\PageIndex{75}$$

$$-9 x-5 \leq-3$$ or $$x+1>3$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{76}$$

$$-5 x-3 \leq 6$$ and $$2 x-1 \geq 6$$

Exercise $$\PageIndex{77}$$

$$-1 \leq-7 x-3 \leq 2$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{78}$$

$$0<5 x-5<9$$

Exercise 79

$$5<9 x-3 \leq 6$$

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 $$\PageIndex{81}$$

$$-2<-7 x+6<6$$

Add texts here. Do not delete this text first.

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 $$\PageIndex{83}$$

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

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{84}$$

$$-\frac{1}{5}<\frac{x}{2}-\frac{1}{4}<\frac{1}{5}$$

Exercise $$\PageIndex{85}$$

$$-\frac{1}{2}<\frac{1}{3}-\frac{x}{2}<\frac{1}{2}$$

Add texts here. Do not delete this text first.

Exercise 86

$$-\frac{2}{3} \leq \frac{1}{2}-\frac{x}{5} \leq \frac{2}{3}$$

Exercise $$\PageIndex{87}$$

$$-1<x-\frac{x+1}{5}<2$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{88}$$

$$-2<x-\frac{2 x-1}{3}<4$$

Exercise $$\PageIndex{89}$$

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

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{90}$$

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

Exercise $$\PageIndex{91}$$

$$x<4-x<5$$

Add texts here. Do not delete this text first.

Exercise $$\PageIndex{92}$$

$$-x<2 x+3 \leq 7$$

Exercise $$\PageIndex{93}$$

$$-x<x+5 \leq 11$$

Exercise $$\PageIndex{94}$$
$$−2x < 3 − x \leq 8$$