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Answer Key 4.2

  1. \(\begin{array}{rrrrrrrr}
    \\ \\
    (3)&(\dfrac{n}{3}&<&3)&\text{or}&\dfrac{-5n}{-5}&<&\dfrac{-10}{-5} \\ \\
    &n&<&9&\text{or}&n&>&2
    \end{array}\)\(\text{Interval notation: } (-\infty, \infty)\)  Number line with positive infinity and negative inifinty
  2. \(\begin{array}{rrrrrrrrr}
    \\ \\
    \dfrac{6m}{6}&\ge & \dfrac{-24}{6}& \text{or}&m&-&7&<&-12 \\
    &&&&&+&7&<&+7 \\
    \hline
    m&\ge & -4& \text{or}&&&m&<&-5
    \end{array}\)\(\text{Interval notation: } (-\infty, -5) \cup [-4, \infty)\)  Numberline with (-5, infinity), (-4, infinity)
  3. \(\begin{array}{rrrrrrrrr}
    \\ \\
    x&+&7& \ge& 12& \text{or}& \dfrac{9x}{9}&<&\dfrac{-45}{9} \\
    &-&7&&-7&&&& \\
    \hline
    &&x& \ge&5& \text{or}&x&<&-5
    \end{array}\)\(\text{Interval notation: } (-\infty, -5) \cup [5, \infty)\) Numberline (-5, infinity) ((5, positive infinity)
  4. \(\begin{array}{rrrrrrrrr}
    \\ \\
    \dfrac{10r}{10}&>&\dfrac{0}{10}&\text{or}&r&-&5&<&-12 \\
    &&&&&+&5&&+5 \\
    \hline
    r&>&0&\text{or}&&&r&<&-7 \\
    \end{array}\)\(\text{Interval notation: } (-\infty, -7) \cup (0, \infty)\)  Number line with (- infinity, -7), (0, positive infinity)
  5. \(\begin{array}{rrrrrrrrr}
    \\ \\
    x&-&6&<&-13&\text{or}&\dfrac{6x}{6}&<&\dfrac{-60}{6} \\
    &+&6&&+6&&&& \\
    \hline
    &&x&<&-7&&x&<&-10 \\
    \end{array}\)\(\text{Interval notation: } (-\infty, -7)\) Numberline (- 7, - infinity)
  6. \(\begin{array}{rrrrrrrrr}
    \\ \\
    9&+&n&<&2&\text{or}&\dfrac{5n}{5}&>&\dfrac{40}{5} \\
    -9&&&&-9&&&& \\
    \hline
    &&n&<&-7&\text{or}&n&>&8 \\
    \end{array}\)\(\text{Interval notation: } (-\infty, -7)\cup (8, \infty)\) Numberline (- infinity, -7) or (8, positive infinity)
  7. \(\begin{array}{rrrrcrrrrr}
    \\ \\
    (8)&(\dfrac{v}{8}&>&-1)&\text{and}&v&-&2&<&1 \\
    &&&&&&+&2&&+2 \\
    \hline
    &v&>&-8&\text{and}&&&v&<&3 \\ \\
    &&-8&<&v&<&3&&&
    \end{array}\)\(\text{Interval notation: } (-8, 3)\) Numberline (-8,3)
  8. \(\begin{array}{rrrcrrrr}
    \\ \\ \\
    \dfrac{-9x}{-9}&<&\dfrac{63}{-9}&\text{and}&(\dfrac{x}{4}&<&1)&(4) \\ \\
    x&>&-7&\text{and}&x&<&4& \\ \\
    &-7&<&x&<&4&&
    \end{array}\)\(\text{Interval notation: } (-7, 4)\)  Numberline (-7,4)
  9. \(\begin{array}{rrrrrrrrr}
    \\ \\
    -8&+&b&<&-3&\text{and}&\dfrac{4b}{4}&<&\dfrac{20}{4} \\
    +8&&&&+8&&&& \\
    \hline
    &&b&<&5&&b&<&5 \\
    \end{array}\)\(\text{Interval notation: } (-\infty, 5)\) Numberline (- infinity, 5)
  10. \(\begin{array}{rrrcrrrr}
    \\ \\ \\ \\
    \dfrac{-6n}{-6}&<&\dfrac{12}{-6}&\text{and}&(\dfrac{n}{3}&<&2)&(3) \\ \\
    n&>&-2&\text{and}&n&<&6& \\ \\
    &-2&<&n&<&6&&
    \end{array}\)\(\text{Interval notation: } (-2, 6)\) Numberline (-2, 6)
  11. \(\begin{array}{rrrrrcrrr}
    \\ \\ \\
    a&+&10&\ge &3&\text{and}&\dfrac{8a}{8}&<&\dfrac{48}{8} \\
    &-&10&&-10&&&& \\
    \hline
    &&a&\ge &-7&\text{and}&a&<&6 \\ \\
    &&&-7&\le &a&<&6&
    \end{array}\)\(\text{Interval notation: } [-7, 6)\) Numberline (-7,6)
  12. \(\begin{array}{rrrrrcrrr}
    \\ \\
    -6&+&v&\ge &0&\text{and}&\dfrac{2v}{2}&>&\dfrac{4}{2} \\
    +6&&&&+6&&&& \\
    \hline
    &&v&\ge &6&\text{and}&v&>&2
    \end{array}\)\(\text{Interval notation: } [6, \infty)\) Numberline [6, positive infinity)
  13. \(\begin{array}{rrrcrrr}
    \\ \\
    3&<&9&+&x&<&7 \\
    -9&&-9&&&&-9 \\
    \hline
    -6&<&&x&&<&-2 \\
    \end{array}\)\(\text{Interval notation: } (-6, -2)\) Numberline (-6,-2)
  14. \(\begin{array}{rrrrrr}
    \\ \\
    (0&\ge & \dfrac{x}{9} & \ge & -1)&(9) \\ \\
    0&\ge & x& \ge & -9& \\ \\
    \end{array}\)\(\text{Interval notation: } [-9, 0]\) Numberline [-9,0]
  15. \(\begin{array}{rrrcrrr}
    \\ \\ \\
    11&<&8&+&k&<&12 \\
    -8&&-8&&&&-8 \\
    \hline
    3&<&&k&&<&4 \\ \\
    \end{array}\)\(\text{Interval notation: } (3, 4)\) Numberline (3,4)
  16. \(\begin{array}{rrrcrrr}
    \\ \\ \\
    -11&<&n&-&9&<&-5 \\
    +9&&&+&9&&+9 \\
    \hline
    -2&<&&n&&<&4 \\ \\
    \end{array}\)\(\text{Interval notation: } (-2, 4)\) Numberline (-2,4)
  17. \(\begin{array}{rrrcrrr}
    \\ \\ \\
    -3&<&x&-&1&<&1 \\
    +1&&&+&1&&+1 \\
    \hline
    -2&<&&x&&<&2 \\ \\
    \end{array}\)\(\text{Interval notation: } (-2, 2)\) Numberline (-2,2)
  18. \(\begin{array}{rrrrrr}
    \\ \\ \\
    (-1&< & \dfrac{p}{8} & <& 0)&(8) \\ \\
    -8&< & p& < & 0& \\ \\
    \end{array}\)\(\text{Interval notation: } (-8, 0)\) Numberline (-8,0)
  19. \(\begin{array}{rrrcrrr}
    \\ \\ \\ \\ \\
    -4&<&8&-&3m&<&11 \\
    -8&&-8&&&&-8 \\
    \hline
    \dfrac{-12}{-3}&<&&\dfrac{-3m}{-3}&&<&\dfrac{3}{-3} \\ \\
    4&>&&m&&>&-1
    \end{array}\)\(\text{Interval notation: } (-1, 4)\)  Numberline (-1,4)
  20. \(\begin{array}{rrrrrcrrrrr}
    \\ \\ \\ \\ \\
    3&+&7r&>&59&\text{or} &-6r&-&3&>&33 \\
    -3&&&&-3&&&+&3&>&+3 \\
    \hline
    &&\dfrac{7r}{7}&>&\dfrac{56}{7}&\text{or} &&&\dfrac{-6r}{-6}&>&\dfrac{36}{-6} \\ \\
    &&r&>&8&\text{or} &&&r&<&-6
    \end{array}\)\(\text{Interval notation: } (-\infty, -6) \cup (8, \infty)\) Numberline (- infinity, -6) or (6, inifinity)
  21. \(\begin{array}{rrrcrrr}
    \\ \\ \\ \\ \\
    -16&<&2n&-&10&<&-2 \\
    +10&&&+&10&&+10 \\
    \hline
    \dfrac{-6}{2}&<&&\dfrac{2n}{2}&&<&\dfrac{8}{2} \\ \\
    -3&<&&n&&<&4
    \end{array}\)\(\text{Interval notation: } (-3, 4)\) Numberline (-3,4)
  22. \(\begin{array}{rrrrrcrrrrr}
    \\ \\ \\ \\ \\
    -6&-&8x&\ge &-6&\text{or} &2&+&10x&>&82 \\
    +6&&&&+6&&-2&&&>&-2 \\
    \hline
    &&\dfrac{-8x}{-8}&\ge&\dfrac{0}{-8}&&&&\dfrac{10x}{10}&>&\dfrac{80}{10} \\ \\
    &&x&\le &0&&\text{or}&&x&>&8
    \end{array}\)\(\text{Interval notation: } (-\infty, 0] \cup (8, \infty)\) Numberline (- infinity,0) or (8, infinity)
  23. \(\begin{array}{rrrrrcrrrrr}
    \\ \\ \\ \\ \\
    -5b&+&10&<&30&\text{and} &7b&+&2&<&-40 \\
    &&-10&&-10&&&-&2&&-2 \\
    \hline
    &&\dfrac{-5b}{-5}&<&\dfrac{20}{-5}&&&&\dfrac{7b}{7}&<&\dfrac{-42}{7} \\ \\
    &&b&>&-4&&\text{and}&&b&<&-6
    \end{array}\)∴ \(\text{No solution}\) No solution
  24. \(\begin{array}{rrrrrcrrrrr}
    \\ \\ \\ \\ \\
    n&+&10&\ge &15&\text{or} &4n&-&5&<&-1 \\
    &-&10&&-10&&&+&5&&+5 \\
    \hline
    &&n&\ge&5&&&&\dfrac{4n}{4}&<&\dfrac{4}{4} \\ \\
    &&n&\ge &5&&\text{or}&&n&<&1
    \end{array}\)\(\text{Interval notation: } (-\infty, 1) \cup [5, \infty)\)  Numberline (negative infinity, 1) or (5 positive inifinity)
  25. \(\begin{array}{rrrrrrrcrrrrrrr}
    \\ \\ \\ \\ \\
    3x&-&9&<&2x&+&10&\text{and}&5&+&7x&<&10x&-&10 \\
    -2x&+&9&&-2x&+&9&&-5&-&10x&&-10x&-&5 \\
    \hline
    &&x&<&19&&&&&&\dfrac{-3x}{-3}&<&\dfrac{-15}{-3}&& \\ \\
    &&x&<&19&&&\text{and}&&&x&>&5&& \\ \\
    &&&&&5&<&x&<&19&&&&&
    \end{array}\)\(\text{Interval notation: } (5, 19)\) Numberline(5,19)
  26. \(\begin{array}{rrrrrrrcrrrrrrr}
    \\ \\ \\
    4n&+&8&<&3n&-&6&\text{or}&10n&-&8&\ge &9&+&9n \\
    -3n&-&8&&-3n&-&8&&-9n&+&8&&+8&-&9n \\
    \hline
    &&n&<&-14&&&\text{or}&&&n&\ge &17&& \\ \\
    \end{array}\)\(\text{Interval notation: } (-\infty, -14) \cup [17, \infty)\) Numerline (- infinity, -14), or (16, positive infinity)
  27. \(\begin{array}{rrrrrrrcrrrrrrr}
    \\ \\ \\ \\ \\
    -8&-&6v&<&8&-&8v&\text{and}&7v&+&9&<&6&+&10v \\
    +8&+&8v&&+8&+&8v&&-10v&-&9&&-9&-&10v \\
    \hline
    &&\dfrac{2v}{2}&<&\dfrac{16}{2}&&&&&&\dfrac{-3v}{-3}&<&\dfrac{-3}{-3}&& \\ \\
    &&v&<&8&&&\text{and}&&&v&>&1&& \\ \\
    &&&&&1&<&v&<&8&&&&&
    \end{array}\)\(\text{Interval notation: } (1, 8)\) Numberline (1,8)
  28. \(\begin{array}{rrrrrrrcrrrrrrr}
    \\ \\ \\ \\ \\
    5&-&2a&\ge &2a&+&1&\text{or}&10a&-&10&\ge &9a&+&9 \\
    -5&-&2a&&-2a&-&5&&-9a&+&10&&-9a&+&10 \\
    \hline
    &&\dfrac{-4a}{-4}&\ge &\dfrac{-4}{-4}&&&&&&a&\ge&19&& \\ \\
    &&a&\le &1&&&\text{or}&&&a&\ge &19&&
    \end{array}\)\(\text{Interval notation: } (-\infty, 1] \cup [19, \infty)\) Numberline (- infinity, 1) or (19, positive infinity)
  29. \(\begin{array}{rrrrrrrcrrrrrrr}
    \\ \\ \\ \\ \\
    1&+&5k&\ge &7k&-&3&\text{or}&k&-&10&>&2k&+&10 \\
    -1&-&7k&&-7k&-&1&&-2k&+&10&&-2k&+&10 \\
    \hline
    &&\dfrac{-2k}{-2}&\ge &\dfrac{-4}{-2}&&&&&&-k&>&20&& \\ \\
    &&k&\le &2&&&\text{or}&&&k&<&-20&&
    \end{array}\)\(\text{Interval notation: } (-\infty, 2]\) Numberline (-2, negative infinity]
  30. \(\begin{array}{rrrrrrrcrrrrrrl}
    \\ \\ \\
    8&-&10r&<&8&+&4r&\text{or}&-6&+&8r&<&2&+&8r \\
    -8&-&4r&&-8&-&4r&&+6&-&8r&&+6&-&8r \\
    \hline
    &&\dfrac{-14r}{-14}&<&\dfrac{0}{-14}&&&&&&0&<&8&\leftarrow &\text{This is always true} \\ \\
    &&r&>&0&&&\text{or}&&&r&\in &\mathbb{R} &&
    \end{array}\)\(\text{Interval notation: } (-\infty, \infty)\) Numberline (negative infinity, positive infinity)
  31. \(\begin{array}{rrrrrrrcrrrrrrr}
    \\ \\ \\ \\ \\
    2x&+&9&\ge &10x&+&1&\text{and}&3x&-&2&<&7x&+&2 \\
    -10x&-&9&&-10x&-&9&&-7x&+&2&&-7x&+&2 \\
    \hline
    &&\dfrac{-8x}{-8}&\ge &\dfrac{-8}{-8}&&&&&&\dfrac{-4x}{-4}&<&\dfrac{4}{-4}&& \\ \\
    &&x&\le &1&&&\text{and}&&&x&>&-1&& \\ \\
    &&&&&-1&<&x&\le &1&&&&&
    \end{array}\)\(\text{Interval notation: } (-1, 1]\) Numberlin (-1,1)
  32. \(\begin{array}{rrrrrrrcrrrrrrr}
    \\ \\ \\ \\ \\
    -9m&+&2&< &-10&-&6m&\text{or}&-m&+&5&\ge &10&+&4m \\
    +6m&-&2&&-2&+&6m&&-4m&-&5&&-5&-&4m \\
    \hline
    &&\dfrac{-3m}{-3}&<&\dfrac{-12}{-3}&&&&&&\dfrac{-5m}{-5}&\ge &\dfrac{5}{-5}&& \\ \\
    &&m&>&4&&&\text{or}&&&m&\le &-1&&
    \end{array}\)\(\text{Interval notation: } (-\infty, -1] \cup (4, \infty)\) Numberline (- infinity, -1) or (4, infinity)

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