# 2.7E: Exercises for Section 2.7

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

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.

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.

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

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

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

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

9) $$(-\infty, 2)$$

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

11) $$(6, \infty)$$

12) $$(5, \infty)$$

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

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

15) $$[0, \infty)$$

16) $$[7, \infty)$$

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

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

19) $$(-\infty, 3]$$

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

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

21)

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

22)

23)

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

24)

25)

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

26)

27)

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

28)

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

29)

$$(4, \infty)$$

30)

31)

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

32)

33)

$$(-\infty, 5]$$

34)

35)

$$[1, \infty)$$

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) $$x+10<19$$

$$(-\infty, 9)$$

38) $$x+17 \geq 7$$

39) $$4 x<8$$

$$(-\infty, 2)$$

40) $$16 x \geq-2$$

41) $$-2 x \leq-2$$

$$[1, \infty)$$

42) $$-18 x>-20$$

43) $$x-18>-10$$

$$(8, \infty)$$

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

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

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

47) $$16 x-6 \leq 18$$

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

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

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

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

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

51) $$5 x+18<38$$

$$(-\infty, 4)$$

52) $$9 x+16<79$$

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

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

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

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

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

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

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

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

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

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

$$(5, \infty)$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

83) $$2.2 x+1.9<-2.3$$

$$(-\infty,-21 / 11)$$
84) $$1.6 x+1.2<1.6$$