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Mathematics LibreTexts

2.8E: Exercises for Section 2.8

  • Page ID
    57739
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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

    Screen Shot 2019-07-29 at 10.25.01 PM.png

    Answer

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

    Screen Shot 2019-08-05 at 10.52.32 AM.png

    Set-Builder Notation: \(\{x \, | \, x \geq 1\}\)

    Interval Notation: \([1, \infty)\)

    Exercise 2

    Screen Shot 2019-07-29 at 10.26.55 PM.png

    Exercise 3

    Screen Shot 2019-07-29 at 10.27.31 PM.png

    Answer

    There are no points that are in both intervals (shaded in both), so there is no intersection. Graph of the intersection:

    Screen Shot 2019-08-05 at 10.53.41 AM.png

    no intersection

    Set-Builder Notation: \(\{ }\)

    Interval Notation: none

    Exercise 4

    Screen Shot 2019-07-29 at 10.28.18 PM.png

    Exercise 5

    Screen Shot 2019-07-29 at 10.29.22 PM.png

    Answer

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

    Screen Shot 2019-08-05 at 10.54.24 AM.png

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

    Interval Notation: \([-6,2]\)

    Exercise 6

    Screen Shot 2019-07-29 at 10.30.25 PM.png

    Exercise 7

    Screen Shot 2019-07-29 at 10.31.23 PM.png

    Answer

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

    Screen Shot 2019-08-05 at 10.55.19 AM.png

    Set-Builder Notation: \(\{x \, | \, x \geq 9\}\)

    Interval Notation: \([9, \infty)\)

    Exercise 8

    Screen Shot 2019-07-29 at 10.33.57 PM.png

     

    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

    Screen Shot 2019-07-29 at 10.37.45 PM.png

    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:

    Screen Shot 2019-08-05 at 10.58.47 AM.png

    Set-Builder Notation: \(\{x \, | \, x \leq-8\}\)

    Interval Notation: \((-\infty,-8]\)

    Exercise 10

    Screen Shot 2019-07-29 at 10.39.08 PM.png

    Exercise 11

    Screen Shot 2019-07-29 at 10.39.38 PM.png

    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:

    Screen Shot 2019-08-05 at 11.01.44 AM.png

    Set-Builder Notation: \(\{x \, | \, x \leq 9 \text { or } x>15\}\)

    Interval Notation: \((-\infty, 9] \cup(15, \infty)\)

    Exercise 12

    Screen Shot 2019-07-29 at 10.40.55 PM.png

    Exercise 13

    Screen Shot 2019-07-29 at 10.42.41 PM.png

    Answer

    The union is the set of all points that are in one interval or the other (shaded in either). Graph of the union:

    Screen Shot 2019-08-05 at 11.03.02 AM.png

    Set-Builder Notation: \(\{x \, | \, x<3\}\)

    Interval Notation: \((-\infty, 3)\)

    Exercise 14

    Screen Shot 2019-07-29 at 10.44.52 PM.png

    Exercise 15

    Screen Shot 2019-07-29 at 10.45.57 PM.png

    Answer

    The union is the set of all points that are in one interval or the other (shaded in either). Graph of the union:

    Screen Shot 2019-08-05 at 11.04.38 AM.png

    Set-Builder Notation: \(\{x \, | \, x \geq 9\}\)

    Interval Notation: \([9, \infty)\)

    Exercise 16

    Screen Shot 2019-07-29 at 10.46.54 PM.png

     

    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:

    Screen Shot 2019-08-05 at 11.06.12 AM.png

    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:

    Screen Shot 2019-08-05 at 11.07.04 AM.png

    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:

    Screen Shot 2019-08-05 at 11.10.10 AM.png

    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:

    Screen Shot 2019-08-05 at 11.12.01 AM.png

    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:

    Screen Shot 2019-08-05 at 11.13.07 AM.png

    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:

    Screen Shot 2019-08-05 at 11.14.04 AM.png

    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:

    Screen Shot 2019-08-05 at 11.14.56 AM.png

    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

    Screen Shot 2019-08-05 at 11.17.54 AM.png

    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

    Screen Shot 2019-08-07 at 10.21.02 PM.png

    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

    Screen Shot 2019-08-08 at 10.56.57 PM.png

    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

    Screen Shot 2019-08-08 at 11.00.01 PM.png

    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

    Screen Shot 2019-08-08 at 11.02.23 PM.png

    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

    Screen Shot 2019-08-08 at 11.04.53 PM.png

    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

    Screen Shot 2019-08-08 at 11.07.23 PM.png

    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

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

    Screen Shot 2019-08-08 at 11.11.56 PM.png

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

    Screen Shot 2019-08-08 at 11.13.59 PM.png

    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

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

    Screen Shot 2019-08-08 at 11.21.51 PM.png

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

    Screen Shot 2019-08-08 at 11.23.30 PM.png

    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 \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    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.

    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

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    Exercise \(\PageIndex{1}\)

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

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    Exercise \(\PageIndex{1}\)

    \(x<4-x<5\)

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{1}\)

    \(−2x < 3 − x \leq 8\)

    Answer

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