{"id":1728,"date":"2021-12-02T19:39:11","date_gmt":"2021-12-03T00:39:11","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-4-2\/"},"modified":"2022-11-02T10:37:21","modified_gmt":"2022-11-02T14:37:21","slug":"answer-key-4-2","status":"publish","type":"back-matter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-4-2\/","title":{"raw":"Answer Key 4.2","rendered":"Answer Key 4.2"},"content":{"raw":"
    \n \t
  1. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrr}\n(3)&(\\dfrac{n}{3}&<&3)&\\text{or}&\\dfrac{-5n}{-5}&<&\\dfrac{-10}{-5} \\\\ \\\\\n&n&<&9&\\text{or}&n&>&2 \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, \\infty)[\/latex]\n\"Number<\/li>\n \t
  2. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrr}\n\\dfrac{6m}{6}&\\ge & \\dfrac{-24}{6}& \\text{or}&m&-&7&<&-12 \\\\\n&&&&&+&7&<&+7 \\\\\n\\hline\nm&\\ge & -4& \\text{or}&&&m&<&-5 \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, -5) \\cup [-4, \\infty)[\/latex]\n\"Numberline<\/li>\n \t
  3. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrr}\nx&+&7& \\ge& 12& \\text{or}& \\dfrac{9x}{9}&<&\\dfrac{-45}{9} \\\\\n&-&7&&-7&&&& \\\\\n\\hline\n&&x& \\ge&5& \\text{or}&x&<&-5 \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\infty, -5) \\cup [5, \\infty)[\/latex]\n\"Numberline<\/li>\n \t
  4. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrr}\n\\dfrac{10r}{10}&>&\\dfrac{0}{10}&\\text{or}&r&-&5&<&-12 \\\\\n&&&&&+&5&&+5 \\\\\n\\hline\nr&>&0&\\text{or}&&&r&<&-7 \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, -7) \\cup (0, \\infty)[\/latex]\n\"Number<\/li>\n \t
  5. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrr}\nx&-&6&<&-13&\\text{or}&\\dfrac{6x}{6}&<&\\dfrac{-60}{6} \\\\\n&+&6&&+6&&&& \\\\\n\\hline\n&&x&<&-7&&x&<&-10 \\\\ \\\\\n\\end{array} \\\\ \\text{Interval notation: } (-\\infty, -7)[\/latex]\n\"Numberline<\/li>\n \t
  6. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrr}\n9&+&n&<&2&\\text{or}&\\dfrac{5n}{5}&>&\\dfrac{40}{5} \\\\\n-9&&&&-9&&&& \\\\\n\\hline\n&&n&<&-7&\\text{or}&n&>&8 \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, -7)\\cup (8, \\infty)[\/latex]\n\"Numberline<\/li>\n \t
  7. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrrr}\n(8)&(\\dfrac{v}{8}&>&-1)&\\text{and}&v&-&2&<&1 \\\\\n&&&&&&+&2&&+2 \\\\\n\\hline\n&v&>&-8&\\text{and}&&&v&<&3 \\\\ \\\\\n&&-8&<&v&<&3&&& \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: }(-8, 3)[\/latex]\n\"Numberline<\/li>\n \t
  8. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrr}\n\\dfrac{-9x}{-9}&<&\\dfrac{63}{-9}&\\text{and}&(\\dfrac{x}{4}&<&1)&(4) \\\\ \\\\\nx&>&-7&\\text{and}&x&<&4& \\\\ \\\\\n&-7&<&x&<&4&& \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-7, 4)[\/latex]\n\"Numberline<\/li>\n \t
  9. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrr}\n-8&+&b&<&-3&\\text{and}&\\dfrac{4b}{4}&<&\\dfrac{20}{4} \\\\\n+8&&&&+8&&&& \\\\\n\\hline\n&&b&<&5&&b&<&5 \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, 5)[\/latex]\n\"Numberline<\/li>\n \t
  10. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrr}\n\\dfrac{-6n}{-6}&<&\\dfrac{12}{-6}&\\text{and}&(\\dfrac{n}{3}&<&2)&(3) \\\\ \\\\\nn&>&-2&\\text{and}&n&<&6& \\\\ \\\\\n&-2&<&n&<&6&& \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-2, 6)[\/latex]\n\"Numberline<\/li>\n \t
  11. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrcrrr}\na&+&10&\\ge &3&\\text{and}&\\dfrac{8a}{8}&<&\\dfrac{48}{8} \\\\\n&-&10&&-10&&&& \\\\\n\\hline\n&&a&\\ge &-7&\\text{and}&a&<&6 \\\\ \\\\\n&&&-7&\\le &a&<&6&\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } [-7, 6)[\/latex]\n\"Numberline<\/li>\n \t
  12. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrcrrr}\n-6&+&v&\\ge &0&\\text{and}&\\dfrac{2v}{2}&>&\\dfrac{4}{2} \\\\\n+6&&&&+6&&&& \\\\\n\\hline\n&&v&\\ge &6&\\text{and}&v&>&2\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } [6, \\infty)[\/latex]\u00a0\"Numberline<\/li>\n \t
  13. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrr}\n3&<&9&+&x&<&7 \\\\\n-9&&-9&&&&-9 \\\\\n\\hline\n-6&<&&x&&<&-2 \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-6, -2)[\/latex]\n\"Numberline<\/li>\n \t
  14. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrr}\n(0&\\ge & \\dfrac{x}{9} & \\ge & -1)&(9) \\\\ \\\\\n0&\\ge & x& \\ge & -9& \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } [-9, 0][\/latex]\n\"Numberline<\/li>\n \t
  15. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrr}\n11&<&8&+&k&<&12 \\\\\n-8&&-8&&&&-8 \\\\\n\\hline\n3&<&&k&&<&4 \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (3, 4)[\/latex]\n\"Numberline<\/li>\n \t
  16. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrr}\n-11&<&n&-&9&<&-5 \\\\\n+9&&&+&9&&+9 \\\\\n\\hline\n-2&<&&n&&<&4 \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-2, 4)[\/latex]\n\"Numberline<\/li>\n \t
  17. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrr}\n-3&<&x&-&1&<&1 \\\\\n+1&&&+&1&&+1 \\\\\n\\hline\n-2&<&&x&&<&2 \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-2, 2)[\/latex]\n\"Numberline<\/li>\n \t
  18. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrr}\n(-1&< & \\dfrac{p}{8} & <& 0)&(8) \\\\ \\\\\n-8&< & p& < & 0& \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-8, 0)[\/latex]\n\"Numberline<\/li>\n \t
  19. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrr}\n-4&<&8&-&3m&<&11 \\\\\n-8&&-8&&&&-8 \\\\\n\\hline\n\\dfrac{-12}{-3}&<&&\\dfrac{-3m}{-3}&&<&\\dfrac{3}{-3} \\\\ \\\\\n4&>&&m&&>&-1\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-1, 4)[\/latex]\n\"Numberline<\/li>\n \t
  20. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrcrrrrr}\n3&+&7r&>&59&\\text{or} &-6r&-&3&>&33 \\\\\n-3&&&&-3&&&+&3&>&+3 \\\\\n\\hline\n&&\\dfrac{7r}{7}&>&\\dfrac{56}{7}&\\text{or} &&&\\dfrac{-6r}{-6}&>&\\dfrac{36}{-6} \\\\ \\\\\n&&r&>&8&\\text{or} &&&r&<&-6\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, -6) \\cup (8, \\infty)[\/latex]\n\"Numberline<\/li>\n \t
  21. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrr}\n-16&<&2n&-&10&<&-2 \\\\\n+10&&&+&10&&+10 \\\\\n\\hline\n\\dfrac{-6}{2}&<&&\\dfrac{2n}{2}&&<&\\dfrac{8}{2} \\\\ \\\\\n-3&<&&n&&<&4 \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-3, 4)[\/latex]\u00a0\"Numberline<\/li>\n \t
  22. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrcrrrrr}\n-6&-&8x&\\ge &-6&\\text{or} &2&+&10x&>&82 \\\\\n+6&&&&+6&&-2&&&>&-2 \\\\\n\\hline\n&&\\dfrac{-8x}{-8}&\\ge&\\dfrac{0}{-8}&&&&\\dfrac{10x}{10}&>&\\dfrac{80}{10} \\\\ \\\\\n&&x&\\le &0&&\\text{or}&&x&>&8\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, 0] \\cup (8, \\infty)[\/latex]\n\"Numberline<\/li>\n \t
  23. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrcrrrrr}\n-5b&+&10&<&30&\\text{and} &7b&+&2&<&-40 \\\\\n&&-10&&-10&&&-&2&&-2 \\\\\n\\hline\n&&\\dfrac{-5b}{-5}&<&\\dfrac{20}{-5}&&&&\\dfrac{7b}{7}&<&\\dfrac{-42}{7} \\\\ \\\\\n&&b&>&-4&&\\text{and}&&b&<&-6\\\\ \\\\\n\\end{array}\\\\ \\therefore \\text{No solution}[\/latex]\n\"No<\/li>\n \t
  24. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrcrrrrr}\nn&+&10&\\ge &15&\\text{or} &4n&-&5&<&-1 \\\\\n&-&10&&-10&&&+&5&&+5 \\\\\n\\hline\n&&n&\\ge&5&&&&\\dfrac{4n}{4}&<&\\dfrac{4}{4} \\\\ \\\\\n&&n&\\ge &5&&\\text{or}&&n&<&1\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, 1) \\cup [5, \\infty)[\/latex]\n\"Numberline<\/li>\n \t
  25. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrcrrrrrrr}\n3x&-&9&<&2x&+&10&\\text{and}&5&+&7x&<&10x&-&10 \\\\\n-2x&+&9&&-2x&+&9&&-5&-&10x&&-10x&-&5 \\\\\n\\hline\n&&x&<&19&&&&&&\\dfrac{-3x}{-3}&<&\\dfrac{-15}{-3}&& \\\\ \\\\\n&&x&<&19&&&\\text{and}&&&x&>&5&& \\\\ \\\\\n&&&&&5&<&x&<&19&&&&&\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (5, 19)[\/latex]\n\"Numberline(5,19)\"<\/li>\n \t
  26. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrcrrrrrrr}\n4n&+&8&<&3n&-&6&\\text{or}&10n&-&8&\\ge &9&+&9n \\\\\n-3n&-&8&&-3n&-&8&&-9n&+&8&&+8&-&9n \\\\\n\\hline\n&&n&<&-14&&&\\text{or}&&&n&\\ge &17&& \\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, -14) \\cup [17, \\infty)[\/latex]\u00a0\"Numerline<\/li>\n \t
  27. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrcrrrrrrr}\n-8&-&6v&<&8&-&8v&\\text{and}&7v&+&9&<&6&+&10v \\\\\n+8&+&8v&&+8&+&8v&&-10v&-&9&&-9&-&10v \\\\\n\\hline\n&&\\dfrac{2v}{2}&<&\\dfrac{16}{2}&&&&&&\\dfrac{-3v}{-3}&<&\\dfrac{-3}{-3}&& \\\\ \\\\\n&&v&<&8&&&\\text{and}&&&v&>&1&& \\\\ \\\\\n&&&&&1&<&v&<&8&&&&&\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (1, 8)[\/latex]\n\"Numberline<\/li>\n \t
  28. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrcrrrrrrr}\n5&-&2a&\\ge &2a&+&1&\\text{or}&10a&-&10&\\ge &9a&+&9 \\\\\n-5&-&2a&&-2a&-&5&&-9a&+&10&&-9a&+&10 \\\\\n\\hline\n&&\\dfrac{-4a}{-4}&\\ge &\\dfrac{-4}{-4}&&&&&&a&\\ge&19&& \\\\ \\\\\n&&a&\\le &1&&&\\text{or}&&&a&\\ge &19&&\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, 1] \\cup [19, \\infty)[\/latex]\u00a0\"Numberline<\/li>\n \t
  29. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrcrrrrrrr}\n1&+&5k&\\ge &7k&-&3&\\text{or}&k&-&10&>&2k&+&10 \\\\\n-1&-&7k&&-7k&-&1&&-2k&+&10&&-2k&+&10 \\\\\n\\hline\n&&\\dfrac{-2k}{-2}&\\ge &\\dfrac{-4}{-2}&&&&&&-k&>&20&& \\\\ \\\\\n&&k&\\le &2&&&\\text{or}&&&k&<&-20&&\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, 2][\/latex]\n\"Numberline<\/li>\n \t
  30. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrcrrrrrrl}\n8&-&10r&<&8&+&4r&\\text{or}&-6&+&8r&<&2&+&8r \\\\\n-8&-&4r&&-8&-&4r&&+6&-&8r&&+6&-&8r \\\\\n\\hline\n&&\\dfrac{-14r}{-14}&<&\\dfrac{0}{-14}&&&&&&0&<&8&\\leftarrow &\\text{This is always true} \\\\ \\\\\n&&r&>&0&&&\\text{or}&&&r&\\in &\\mathbb{R} &&\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, \\infty)[\/latex]\n\"Numberline<\/li>\n \t
  31. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrcrrrrrrr}\n2x&+&9&\\ge &10x&+&1&\\text{and}&3x&-&2&<&7x&+&2 \\\\\n-10x&-&9&&-10x&-&9&&-7x&+&2&&-7x&+&2 \\\\\n\\hline\n&&\\dfrac{-8x}{-8}&\\ge &\\dfrac{-8}{-8}&&&&&&\\dfrac{-4x}{-4}&<&\\dfrac{4}{-4}&& \\\\ \\\\\n&&x&\\le &1&&&\\text{and}&&&x&>&-1&& \\\\ \\\\\n&&&&&-1&<&x&\\le &1&&&&&\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-1, 1][\/latex]\u00a0\"Numberlin<\/li>\n \t
  32. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrcrrrrrrr}\n-9m&+&2&< &-10&-&6m&\\text{or}&-m&+&5&\\ge &10&+&4m \\\\\n+6m&-&2&&-2&+&6m&&-4m&-&5&&-5&-&4m \\\\\n\\hline\n&&\\dfrac{-3m}{-3}&<&\\dfrac{-12}{-3}&&&&&&\\dfrac{-5m}{-5}&\\ge &\\dfrac{5}{-5}&& \\\\ \\\\\n&&m&>&4&&&\\text{or}&&&m&\\le &-1&&\\\\ \\\\\n\\end{array}\\\\ \\text{Interval notation: } (-\\infty, -1] \\cup (4, \\infty)[\/latex]\n\"Numberline<\/li>\n<\/ol>","rendered":"
      \n
    1. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]
      \n\"Number<\/li>\n
    2. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Numberline<\/li>\n
    3. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Numberline<\/li>\n
    4. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Number<\/li>\n
    5. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Numberline<\/li>\n
    6. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]
      \n\"Numberline<\/li>\n
    7. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Numberline<\/li>\n
    8. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Numberline<\/li>\n
    9. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Numberline<\/li>\n
    10. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Numberline<\/li>\n
    11. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Numberline<\/li>\n
    12. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\u00a0\"Numberline<\/li>\n
    13. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrr} 3&<&9&+&x&<&7 \\\\ -9&&-9&&&&-9 \\\\ \\hline -6&<&&x&&<&-2 \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-6, -2)[\/latex]\n\"Numberline<\/li>\n
    14. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrr} (0&\\ge & \\dfrac{x}{9} & \\ge & -1)&(9) \\\\ \\\\ 0&\\ge & x& \\ge & -9& \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } [-9, 0][\/latex]
      \n\"Numberline<\/li>\n
    15. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrr} 11&<&8&+&k&<&12 \\\\ -8&&-8&&&&-8 \\\\ \\hline 3&<&&k&&<&4 \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (3, 4)[\/latex]\n\"Numberline<\/li>\n
    16. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrr} -11&<&n&-&9&<&-5 \\\\ +9&&&+&9&&+9 \\\\ \\hline -2&<&&n&&<&4 \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-2, 4)[\/latex]\n\"Numberline<\/li>\n
    17. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrr} -3&<&x&-&1&<&1 \\\\ +1&&&+&1&&+1 \\\\ \\hline -2&<&&x&&<&2 \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-2, 2)[\/latex]\n\"Numberline<\/li>\n
    18. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrr} (-1&< & \\dfrac{p}{8} & <& 0)&(8) \\\\ \\\\ -8&< & p& < & 0& \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-8, 0)[\/latex]\n\"Numberline<\/li>\n
    19. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]
      \n\"Numberline<\/li>\n
    20. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Numberline<\/li>\n
    21. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\u00a0\"Numberline<\/li>\n
    22. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]
      \n\"Numberline<\/li>\n
    23. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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}\\\\ \\therefore \\text{No solution}[\/latex]\n\"No<\/li>\n
    24. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Numberline<\/li>\n
    25. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Numberline(5,19)\"<\/li>\n
    26. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\u00a0\"Numerline<\/li>\n
    27. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\n\"Numberline<\/li>\n
    28. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]\u00a0\"Numberline<\/li>\n
    29. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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][\/latex]\n\"Numberline<\/li>\n
    30. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]
      \n\"Numberline<\/li>\n
    31. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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][\/latex]\u00a0\"Numberlin<\/li>\n
    32. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{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)[\/latex]
      \n\"Numberline<\/li>\n<\/ol>\n","protected":false},"author":90,"menu_order":31,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"back-matter-type":[],"contributor":[],"license":[56],"class_list":["post-1728","back-matter","type-back-matter","status-publish","hentry","license-cc-by-nc-sa"],"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1728","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":1,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1728\/revisions"}],"predecessor-version":[{"id":1729,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1728\/revisions\/1729"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1728\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=1728"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter-type?post=1728"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=1728"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=1728"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}