2.7E: Exercises for Section 2.7

Last updated
Save as PDF

Page ID: 57426

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\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]{\| #1 \|}$

$ \newcommand{\inner}[2]{\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]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

1) Draw a number line, then plot the numbers $4,3,-4,7 / 8$, and $−8/3$ on your number line. Label each point with its value. Finally, list the numbers in order, from smallest to largest.

Answer: $Ans 2.6.1.png$

2) Draw a number line, then plot the numbers $5,3,-4,5 / 7$, and $−4/3$ on your number line. Label each point with its value. Finally, list the numbers in order, from smallest to largest.

3) Draw a number line, then plot the numbers $-5,5,4,2 / 3$, and $8/3$ on your number line. Label each point with its value. Finally, list the numbers in order, from smallest to largest.

Answer: $Ans 2.6.3.png$

4) Draw a number line, then plot the numbers $-3,-2,4,1 / 3$, and $5/2$ on your number line. Label each point with its value. Finally, list the numbers in order, from smallest to largest.

In Exercises 5-20, shade each of the following sets on a number line.

5) $\{x \, | \, x \geq-7\}$

Answer: $Ans 2.6.5.png$

6) $\{x \, | \, x \geq-1\}$

7) $\{x \, | \, x<2\}$

Answer: $Ans 2.6.7.png$

8) $\{x \, | \, x<-6\}$

9) $(-\infty, 2)$

Answer: $Ans 2.6.9.png$

10) $(-\infty,-9)$

11) $(6, \infty)$

Answer: $Ans 2.6.11.png$

12) $(5, \infty)$

13) $\{x \, | \, x>7\}$

Answer: $Ans 2.6.13.png$

14) $\{x \, | \, x>-8\}$

15) $[0, \infty)$

Answer: $Ans 2.6.15.png$

16) $[7, \infty)$

17) $\{x \, | \, x \leq-2\}$

Answer: $Ans 2.6.17.png$

18) $\{x \, | \, x \leq 7\}$

19) $(-\infty, 3]$

Answer: $Ans 2.6.19.png$

20) $(-\infty,-1]$

In Exercises 21-28, use set-builder notation to describe the shaded region on the given number line.

21)

Answer: $\{x \, | \, x \leq 9\}$

22)

23)

Answer: $\{x \, | \, x<-8\}$

24)

25)

Answer: $\{x \, | \, x>-2\}$

26)

27)

Answer: $\{x \, | \, x \geq-3\}$

28)

In Exercises 29-36, use interval notation to describe the shaded region on the given number line.

29)

Answer: $(4, \infty)$

30)

31)

Answer: $(-\infty,-2)$

32)

33)

Answer: $(-\infty, 5]$

34)

35)

Answer: $[1, \infty)$

36)

In Exercises 37-44, solve each of the given inequalities. Sketch the solution on a number line, then use set-builder and interval notation to describe your solution.

37) $x+10<19$

Answer: $(-\infty, 9)$

38) $x+17 \geq 7$

39) $4 x<8$

Answer: $(-\infty, 2)$

40) $16 x \geq-2$

41) $-2 x \leq-2$

Answer: $[1, \infty)$

42) $-18 x>-20$

43) $x-18>-10$

Answer: $(8, \infty)$

44) $x-8 \leq-18$

In Exercises 45-62, solve each of the given inequalities. Sketch the solution on a number line, then use set-builder and interval notation to describe your solution.

45) $-5 x-6 \geq 4-9 x$

Answer: $[5 / 2, \infty)$

46) $2 x-7 \geq-3-4 x$

47) $16 x-6 \leq 18$

Answer: $(-\infty, 3 / 2]$

48) $8 x-14 \leq-12$

49) $-14 x-6 \geq-10-4 x$

Answer: $(-\infty, 2 / 5]$

50) $-13 x-4 \geq-2-5 x$

51) $5 x+18<38$

Answer: $(-\infty, 4)$

52) $9 x+16<79$

53) $-16 x-5 \geq-11-6 x$

Answer: $(-\infty, 3 / 5]$

54) $-11 x-7 \geq-15-5 x$

55) $2 x-9 \geq 5-8 x$

Answer: $[7 / 5, \infty)$

56) $-3 x-6 \geq-2-9 x$

57) $-10 x-4 \leq 18$

Answer: $[-11 / 5, \infty)$

58) $-6 x-14 \leq 1$

59) $-12 x+4<-56$

Answer: $(5, \infty)$

60) $-18 x+6<-12$

61) $15 x+5<6 x+2$

Answer: $(-\infty,-1 / 3)$

62) $12 x+8<3 x+5$

In Exercises 63-76, solve each of the given inequalities. Sketch the solution on a number line, then use set-builder and interval notation describe your solution.

63) $\dfrac{3}{2} x>\dfrac{9}{8}$

Answer: $(3 / 4, \infty)$

64) $\dfrac{6}{7} x>\dfrac{3}{4}$

65) $x+\dfrac{3}{2}<\dfrac{9}{5}$

Answer: $(-\infty, 3 / 10)$

66) $x+\dfrac{1}{4}<-\dfrac{1}{5}$

67) $\dfrac{4}{7}-\dfrac{1}{6} x \leq \dfrac{4}{3} x-\dfrac{1}{2}$

Answer: $[5 / 7, \infty)$

68) $\dfrac{5}{3}-\dfrac{3}{4} x \leq \dfrac{7}{4} x-\dfrac{3}{5}$

69) $x-\dfrac{3}{8} \geq-\dfrac{9}{7}$

Answer: $[-51 / 56, \infty)$

70) $x-\dfrac{7}{2} \geq \dfrac{1}{5}$

71) $\dfrac{6}{5} x \leq-\dfrac{4}{7}$

Answer: $[10 / 21, \infty)$

72) $\dfrac{4}{3} x \leq \dfrac{2}{9}$

73) $-\dfrac{6}{5} x-\dfrac{7}{3} \leq \dfrac{5}{9}-\dfrac{2}{9} x$

Answer: $[-65 / 22, \infty)$

74) $-\dfrac{3}{7} x-\dfrac{1}{2} \leq \dfrac{3}{2}-\dfrac{2}{7} x$

75) $\dfrac{9}{7} x+\dfrac{9}{2}>\dfrac{1}{7} x+\dfrac{7}{2}$

Answer: $(-7 / 8, \infty)$

76) $\dfrac{5}{7} x+\dfrac{9}{2}>\dfrac{1}{3} x+\dfrac{5}{2}$

In Exercises 77-84, solve each of the given inequalities. Sketch the solution on a number line, then use set-builder and interval notation containing fractions in reduced form to describe your solution.

77) $-3.7 x-1.98 \leq 3.2$

Answer: $[-7 / 5, \infty)$

78) $-3.6 x-3.32 \leq 0.8$

79) $-3.4 x+3.5 \geq 0.9-2.2 x$

Answer: $(-\infty, 13 / 6]$

80) $-2.6 x+3.1 \geq-2.9-1.7 x$

81) $-1.3 x+2.9>-2.6-3.3 x$

Answer: $(-11 / 4, \infty)$

82) $2.5 x+2.1>1.4-3.8 x$

83) $2.2 x+1.9<-2.3$

Answer: $(-\infty,-21 / 11)$

84) $1.6 x+1.2<1.6$

Contributors and Attributions

David Arnold (Retired Professor (Mathematics) at College of the Redwoods)

Search

Text Color

Text Size

Margin Size

Font Type