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Mathematics LibreTexts

2.7E: Exercises for Section 2.7

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    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)

    Exercise 2.6.21.png
    Answer

    \(\{x \, | \, x \leq 9\}\)

    22)

    Exercise 2.6.22.png

    23)

    Exercise 2.6.23.png
    Answer

    \(\{x \, | \, x<-8\}\)

    24)

    Exercise 2.6.24.png

    25)

    Exercise 2.6.25.png
    Answer

    \(\{x \, | \, x>-2\}\)

    26)

    Exercise 2.6.26.png

    27)

    Exercise 2.6.27.png
    Answer

    \(\{x \, | \, x \geq-3\}\)

    28)

    Exercise 2.6.28.png

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

    29)

    Exercise 2.6.29.png
    Answer

    \((4, \infty)\)

    30)

    Exercise 2.6.30.png

    31)

    Exercise 2.6.31.png
    Answer

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

    32)

    Exercise 2.6.32.png

    33)

    Exercise 2.6.33.png
     
    Answer

    \((-\infty, 5]\)

    34)

    Exercise 2.6.34.png

    35)

    Exercise 2.6.35.png
    Answer

    \([1, \infty)\)

    36)

    Exercise 2.6.36.png

    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) \(1.6 x+1.2<1.6\)

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