2.8E: Exercises for Section 2.8
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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: \(\{x \, | \, x \geq 1\}\)
Interval Notation: \([1, \infty)\)
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: \(\{x \, | \, -6 \leq x \leq 2\}\)
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: \(\{x \, | \, x \geq 9\}\)
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
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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: \(\{x \, | \, x \leq-8\}\)
Interval Notation: \((-\infty,-8]\)
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: \(\{x \, | \, x \leq 9 \text { or } x>15\}\)
Interval Notation: \((-\infty, 9] \cup(15, \infty)\)
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: \((-\infty, 3)\)
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: \(\{x \, | \, x \geq 9\}\)
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
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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
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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 \((-\infty, \infty)\). Graph of the set:
Exercise 20
\(\{x \, | \, x>-7 \text { and } x>-4\}\)
Exercise 21
\(\{x \, | \, x \geq -1 \text { or } x>6\}\)
- Answer
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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
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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
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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
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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
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There are no numbers that satisfy both inequalities. Thus, there is no intersection. Graph of the set:
Exercise 30
\(\{x \, | \, x<5 \text { or } x<10\}\)
Exercise 31
\(\{x \, | \, x \leq 5 \text { and } x \geq-1\}\)
- Answer
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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. Answers include worked out solutions.
Exercise 33
\(-8 x-3 \leq-16 x-1\)
- Answer
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\[\begin{aligned} & -8 x-3 \leq-16 x-1 \\ \Longrightarrow \quad & − 8x + 16x \leq −1 + 3 \\ \Longrightarrow \quad& 8x \leq 2 \\ \Longrightarrow \quad & x \leq \frac{1}{4}\end{aligned}\]
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
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\[\begin{aligned} & -12 x+5 \leq-3 x-4 \\ \Longrightarrow \quad & -12x + 3x \leq −4 − 5 \\ \Longrightarrow \quad& -9x \leq -9 \\ \Longrightarrow \quad & x \geq 1\end{aligned}\]
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
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\[\begin{aligned} & − 11x − 9 < −3x + 1 \\ \Longrightarrow \quad & − 11x + 3x < 1 + 9 \\ \Longrightarrow \quad& − 8x < 10 \\ \Longrightarrow \quad & x > -\frac{5}{4}\end{aligned}\]
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
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\[\begin{align*} & 4x − 5 > 5x − 7\\ \Longrightarrow \quad & 4x − 5x > −7 + 5 \\ \Longrightarrow \quad& − x > −2 \\ \Longrightarrow \quad &x < 2\end{align*}\]
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
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\[\begin{aligned} & 2x − 1 > 7x + 2\\ \Longrightarrow \quad & 2x − 7x > 2 + 1 \\ \Longrightarrow \quad& − 5x > 3 \\ \Longrightarrow \quad &x < −\frac{3}{5}\end{aligned}\]
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
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\[\begin{aligned} & − 3x + 3 < −11x − 3\\ \Longrightarrow \quad & − 3x + 11x < −3 − 3 \\ \Longrightarrow \quad& 8x < −6 \\ \Longrightarrow \quad &x < -\frac{3}{4}\end{aligned}\]
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. 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
\(2 x-1<4\) or \(7 x+1 \geq-4\)
- Answer
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\[\begin{aligned} & 2x − 1 < 4 \text{ or } 7x + 1 \geq −4\\ \Longrightarrow \quad & 2x < 5\quad \text{or}\quad 7x \geq −5 \\ \Longrightarrow \quad&x<\frac{5}{2}\quad\text{or}\quad x\geq-\frac{5}{7}\end{aligned}\]
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
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\[\begin{aligned} & − 6x − 4 < −4 \text{ and } − 3x + 7 \geq −5\\ \Longrightarrow \quad & -6x < 0\quad \text{and}\quad -3x \geq −12 \\ \Longrightarrow \quad&x>0\quad\text{and}\quad x\leq4 \\ \Longrightarrow \quad & 0< x \leq 4 \end{aligned}\]
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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No Solution
Set-Builder Notation: \( \{ \; \} \)
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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No Solution
Set-Builder Notation: \( \{ \; \} \)
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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Interval Notation: \( \left( -\dfrac{1}{5}, 2 \right] \)
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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Interval Notation: \((−\infty, 3) \)
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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Interval Notation: \( \left(−\infty, -\dfrac{8}{9} \right) \)
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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Add texts here. \( (-\infty,1] \cup [4, \infty) \)
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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Interval Notation: \( \left( -\dfrac{13}{3}, -\dfrac{7}{5} \right) \)
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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Interval Notation: \( [ 1, \infty) \)
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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No Solution
Set-Builder Notation: \( \{ \; \} \)
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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No Solution
Set-Builder Notation: \( \{ \; \} \)
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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\( (0, 3] \)
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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\((−\infty, \infty) \)
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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\((−\infty, \infty) \)
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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\( \left(-\dfrac{2}{9}, \infty \right) \)
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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Interval Notation: \( \left[ -\dfrac{5}{7}, -\dfrac{2}{7} \right] \)
Exercise \(\PageIndex{78}\)
\(0<5 x-5<9\)
Exercise 79
\(5<9 x-3 \leq 6\)
- Answer
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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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Interval Notation: \( \left( 0, \dfrac{8}{7} \right) \)
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. Only the interval notation version of the answer is given for Answers in #83-94.
Exercise \(\PageIndex{83}\)
\(-\frac{1}{3}<\frac{x}{2}+\frac{1}{4}<\frac{1}{3}\)
- Answer
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Interval Notation: \( \left( -\dfrac{7}{6}, \dfrac{1}{6} \right) \)
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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Interval Notation: \( \left( -\dfrac{1}{3}, \dfrac{5}{3} \right) \)
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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Interval Notation: \( \left( -1, \dfrac{11}{4} \right) \)
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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Interval Notation: \( \left( -13, 11 \right] \)
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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Interval Notation: \( \left( -1, 2 \right) \)
Exercise \(\PageIndex{92}\)
\(-x<2 x+3 \leq 7\)
Exercise \(\PageIndex{93}\)
\(-x<x+5 \leq 11\)
- Answer
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Interval Notation: \( \left( -\dfrac{5}{2}, 6 \right] \)
Exercise \(\PageIndex{94}\)
\(−2x < 3 − x \leq 8\)
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. |
x
- Answer
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1. (a) \(\{x| \, 3 \leq x \leq 4\}\) (b) \([3,4]\)
3. (a) \(\{x| -4 \lt x \lt 1\}\) (b) \((-4,1)\)5. (a) \(\{x| -1 \leq x \leq 5\}\) (b) \([-1,5]\)
7. (a) \(\{x| -1 \lt x \leq 5\}\) (b) \((-1,5]\)