{"id":1763,"date":"2021-12-02T19:39:20","date_gmt":"2021-12-03T00:39:20","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-4-3\/"},"modified":"2022-11-02T10:37:28","modified_gmt":"2022-11-02T14:37:28","slug":"answer-key-4-3","status":"publish","type":"back-matter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-4-3\/","title":{"raw":"Answer Key 4.3","rendered":"Answer Key 4.3"},"content":{"raw":"
    \n \t
  1. [latex]\\begin{array}{rrcrrr}\n-3& <& x& <& 3& \\hspace{0.25in} \\text{Interval notation: } (-3,3)\n\\end{array}[\/latex]\n\"(-3,3)\"<\/li>\n \t
  2. [latex]\\begin{array}{rrcrrr}\n-8& \\le& x& \\le& 8 & \\hspace{0.25in} \\text{Interval notation: } [-8,8]\n\\end{array}[\/latex]\n\"Numberline<\/li>\n \t
  3. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrcrrr}\n\\dfrac{-6}{2}&<&\\dfrac{2x}{2}&<&\\dfrac{6}{2}& \\\\ \\\\\n-3&<&x&<&3& \\hspace{0.25in} \\text{Interval notation: } (-3,3)\n\\end{array}[\/latex]\n\"Numerline<\/li>\n \t
  4. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrr}\n-4&<&x&+&3&<&4& \\\\\n-3&&&-&3&&-3& \\\\\n\\hline\n-7&<&&x&&<&1& \\hspace{0.25in} \\\\ \\\\\n\\end{array}\\\\\n\\text{Interval notation: } (-7,1)[\/latex]\n\"Numberline<\/li>\n \t
  5. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrr}\n-6&<&x&-&2&<&6& \\\\\n+2&&&+&2&&+2& \\\\\n\\hline\n-4&<&&x&&<&8& \\hspace{0.25in}\\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } (-4,8)\n[\/latex]\n\"Numberline<\/li>\n \t
  6. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrr}-12&<&x&-&8&<&12& \\\\\n+8&&&+&8&&+8& \\\\\n\\hline\n-4&<&&x&&<&20& \\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } (-4,20)\n[\/latex]\n\"Numberline<\/li>\n \t
  7. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrr}\n-3&<&x&-&7&<&3& \\\\\n+7&&&+&7&&+7& \\\\\n\\hline\n4&<&&x&&<&10& \\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } (4,10)\n[\/latex]\n\"Numberline<\/li>\n \t
  8. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrr}\n-4&\\le &x&+&3&\\le &4& \\\\\n-3&&&-&3&&-3& \\\\\n\\hline\n-7&\\le &&x&&\\le &1& \\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } [-7,1]\n[\/latex]\n\"Numberline<\/li>\n \t
  9. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrr}\n-9&<&3x&-&2&<&9& \\\\\n+2&&&+&2&&+2& \\\\\n\\hline\n\\dfrac{-7}{3}&<&&\\dfrac{3x}{3}&&<&\\dfrac{11}{3}&\\\\ \\\\\n-\\dfrac{7}{3}&<&&x&&<&\\dfrac{11}{3}& \\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } (-\\dfrac{7}{3},\\dfrac{11}{3})\n[\/latex]\n\"Numberline<\/li>\n \t
  10. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrr}\n-9&<&2x&+&5&<&9& \\\\\n-5&&&-&5&&-5& \\\\\n\\hline\n\\dfrac{-14}{2}&<&&\\dfrac{2x}{2}&&<&\\dfrac{4}{2}&\\\\ \\\\\n-7&<&&x&&<&2& \\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } (-7,2)[\/latex]\n\"Numberline<\/li>\n \t
  11. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrr}\n1&+&2|x&-&1|&\\le & 9 \\\\\n-1&&&&&&-1 \\\\\n\\hline\n&&\\dfrac{2}{2}|x&-&1|&\\le & \\dfrac{8}{2} \\\\ \\\\\n&&|x&-&1|&\\le & 4 \\\\\n-4&\\le &x&-&1&\\le & 4 \\\\\n+1&&&+&1&\\le &+1 \\\\\n\\hline\n-3&\\le &&x&&\\le &5& \\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } [-3,5]\n[\/latex]\n\"Numberline<\/li>\n \t
  12. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrr}\n10&-&3|x&-&2|&\\ge & 4 \\\\\n-10&&&&&&-10 \\\\\n\\hline\n&&\\dfrac{-3}{-3}|x&-&2|&\\ge & \\dfrac{-6}{-3} \\\\ \\\\\n&&|x&-&2|&\\le & 2 \\\\\n-2&\\le &x&-&2&\\le & 2 \\\\\n+2&&&+&2&&+2 \\\\\n\\hline\n0&\\le &&x&&\\le &4& \\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } [0,4]\n[\/latex]\n\"Numberline<\/li>\n \t
  13. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrl}\n6&-&|2x&-&5|&>&3& \\\\\n-6&&&&&&-6& \\\\\n\\hline\n&&(-|2x&-&5|&>&-3)&(-1) \\\\\n&&|2x&-&5|&<&3& \\\\\n-3&<&2x&-&5&<&3& \\\\\n+5&&&+&5&&+5& \\\\\n\\hline\n\\dfrac{2}{2}&<&&\\dfrac{2x}{2}&&<&\\dfrac{8}{2}& \\\\ \\\\\n1&<&&x&&<&4& \\\\ \\\\ \\end{array} \\\\\n\\text{Interval notation: } (1,4)\n[\/latex]\n\"Numberline<\/li>\n \t
  14. [latex]\\begin{array}{rrcrrrrrr}\nx& <&-5&\\text{or}&5&<&x& \\hspace{0.25in} \\text{Interval notation: } (-\\infty,-5)\\cup (5, \\infty)\n\\end{array}[\/latex]\n\"Numberline<\/li>\n \t
  15. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrr}\n\\dfrac{3x}{3}&<&\\dfrac{-5}{3}&\\text{or}&\\dfrac{5}{3}&<&\\dfrac{3x}{3}& \\\\ \\\\\nx&<&-\\dfrac{5}{3}&\\text{or}&\\dfrac{5}{3}&<&x&\\\\ \\\\ \\end{array} \\\\\n\\text{Interval notation: } (-\\infty, -\\dfrac{5}{3})\\cup (\\dfrac{5}{3}, \\infty) \\\\\n[\/latex]\n\"(-<\/li>\n \t
  16. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrrr}\nx&-&4&<&-5&\\text{or}&5&<&x&-&4&\\\\\n&+&4&&+4&\\text{or}&+4&&&+&4&\\\\\n\\hline\n&&x&<&-1&\\text{or}&9&<&x&&&\\\\ \\\\ \\end{array} \\\\\n\\text{Interval notation: } (-\\infty, -1)\\cup (9, \\infty)\n[\/latex]\n\"(-<\/li>\n \t
  17. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrrr}\nx&+&3&<&-3&\\text{or}&3&<&x&+&3& \\\\\n&-&3&&-3&\\text{or}&-3&&&-&3& \\\\\n\\hline\n&&x&<&-6&\\text{or}&0&<&x&&&\\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } (-\\infty, -6)\\cup (0, \\infty)\\\\\n[\/latex]\n\"(-<\/li>\n \t
  18. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrrr}\n2x&-&4&<&-6&\\text{or}&6&<&2x&-&4& \\\\\n&+&4&&+4&&+4&&&+&4& \\\\\n\\hline\n&&\\dfrac{2x}{2}&<&\\dfrac{-2}{2}&&\\dfrac{10}{2}&<&\\dfrac{2x}{2}&&& \\\\ \\\\\n&&x&<&-1&\\text{or}&5&<&x&&&\\\\ \\\\ \\end{array} \\\\ \\text{Interval notation: } (-\\infty, -1)\\cup (5, \\infty)\\\\\n[\/latex]\n\"(negative<\/li>\n \t
  19. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrrr}\nx&-&5&<&-3&\\text{or}&3&<&x&-&5& \\\\\n&+&5&&+5&&+5&&&+&5& \\\\\n\\hline\n&&x&<&2&\\text{or}&8&<&x&&&\\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } (-\\infty, 2)\\cup (8, \\infty)\\\\\n[\/latex] HAVE TO ADD 2 2 etc.\n\"(1,<\/li>\n \t
  20. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrl}\n3&-|2&-&x|&<&1&&&&&\\\\\n-3&&&&&-3&&&&&\\\\\n\\hline\n&(-|2&-&x|&<&-2)&(-1)&&&&\\\\\n&|2&-&x|&>&2&&&&&\\\\ \\\\\n2&-&x&<&-2&\\text{or}&2&<&2&-&x\\\\\n-2&&&&-2&&-2&&-2&&\\\\\n\\hline\n&&-x&<&-4&\\text{or}&0&<&-x&&\\\\\n&&x&>&4&\\text{or}&0&>&x&&\\\\ \\\\ \\end{array} \\\\ \\text{Interval notation: } (-\\infty, 0)\\cup (4, \\infty)\\\\\n[\/latex]\n\"(negative<\/li>\n \t
  21. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrl}\n4&+&3|x&-&1|&<&10&&&&\\\\\n-4&&&&&&-4&&&&\\\\\n\\hline\n&&\\dfrac{3}{3}|x&-&1|&<&\\dfrac{6}{3}&&&&\\\\ \\\\\n&&|x&-&1|&<&2&&&&\\\\ \\\\\nx&-&1&<&-2&\\text{or}&2&<&x&-&1\\\\\n&+&1&&+1&&+1&&&+&1\\\\\n\\hline\n&&x&<&-1&\\text{or}&3&<&x&&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\infty, -1)\\cup (3, \\infty)\\\\\n[\/latex]\n\"(negative<\/li>\n \t
  22. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrl}\n3&-&2|3x&-&1|&\\ge &-7& \\\\\n-3&&&&&&-3& \\\\\n\\hline\n&&\\dfrac{-2}{-2}|3x&-&1|&\\ge &\\dfrac{-10}{-2}& \\\\ \\\\\n&&|3x&-&1|&\\le &5& \\\\ \\\\ \\\\\n-5&\\le &3x&-&1& \\le & 5& \\\\\n+1&&&+&1&&+1& \\\\\n\\hline\n\\dfrac{-4}{3}&\\le &&\\dfrac{3x}{3}&& \\le & \\dfrac{6}{3}& \\\\ \\\\\n-\\dfrac{4}{3}&\\le &&x&& \\le & 2 &\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } [-\\dfrac{4}{3}, 2]\n[\/latex]\n\"(-4\/3,<\/li>\n \t
  23. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrcrrrrl}\n3&-&2|x&-&5|&\\le & -15&&&& \\\\\n-3&&&&&&-3&&&& \\\\\n\\hline\n&&\\dfrac{-2}{-2}|x&-&5|&\\le & \\dfrac{-18}{-2}&&&& \\\\ \\\\\n&&|x&-&5|&\\ge & 9&&&& \\\\ \\\\\nx&-&5&\\le &-9&\\text{or}&9&\\le &x&-&5 \\\\\n&+&5&&+5&&+5&&&+&5 \\\\\n\\hline\n&&x&\\le &-4&\\text{or}&14&\\le &x&&\\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } (-\\infty, -4]\\cup [14, \\infty)\n[\/latex]\n\"(-<\/li>\n \t
  24. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrcrrcrrrll}\n4&-&6|-6&-&3x|&\\le & -5&&&& \\\\\n-4&&&&&&-4&&&& \\\\\n\\hline\n&&\\dfrac{-6}{-6}|-6&-&3x|&\\le &\\dfrac{-9}{-6}&&&& \\\\ \\\\\n&&|-6&-&3x|&\\ge &\\dfrac{3}{2}&&&& \\\\ \\\\ \\\\\n-6&-&3x&\\le &-\\dfrac{3}{2}&\\text{or}&\\dfrac{3}{2}&\\le &-6&-\\phantom{0}3x& \\\\ \\\\\n+6&&&&+6&&+6&&+6&& \\\\\n\\hline\n&&\\dfrac{-3x}{-3}&\\le &\\dfrac{\\dfrac{9}{2}}{-3}&&\\dfrac{\\dfrac{15}{2}}{-3}&\\le &\\dfrac{-3x}{-3}&& \\\\ \\\\\n&&x&\\ge &-\\dfrac{3}{2}&\\text{or}&-\\dfrac{5}{2}&\\ge &x&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\infty, -\\dfrac{5}{2}]\\cup [-\\dfrac{3}{2}, \\infty)\n[\/latex]\n\"[-5\/2,<\/li>\n \t
  25. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrr}\n-2&-&3|4&-&2x|&\\ge & -8& \\\\\n+2&&&&&&+2& \\\\\n\\hline\n&&\\dfrac{-3}{-3}|4&-&2x|&\\ge &\\dfrac{-6}{-3}& \\\\ \\\\\n&&|4&-&2x|&\\le &2& \\\\ \\\\ \\\\\n-2&\\le &4&-&2x&\\le &2& \\\\\n-4&&-4&&&&-4& \\\\\n\\hline\n\\dfrac{-6}{-2}&\\le &&\\dfrac{-2x}{-2}&&\\le &\\dfrac{-2}{-2}& \\\\ \\\\\n3&\\ge &&x&&\\ge &1&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } [1,3]\n[\/latex]\n\"[1,3]\"<\/li>\n \t
  26. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrcrr}\n-3&-&2|4x&-&5|&\\ge & 1& \\\\\n+3&&&&&&+3& \\\\\n\\hline\n&&\\dfrac{-2}{-2}|4x&-&5|&\\ge &\\dfrac{4}{-2}& \\\\ \\\\\n&&|4x&-&5|&\\le &-2& \\\\ \\\\\n&&&&&\\uparrow&& \\\\\n&&&&&\\text{Cannot be true.}&&\\text{No solution.}\n\\end{array}[\/latex]\n\"Numerline<\/li>\n \t
  27. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrrl}\n4&-&5|-2x&-&7|&<&-1&&&&& \\\\\n-4&&&&&&-4&&&&& \\\\\n\\hline\n&&\\dfrac{-5}{-5}|-2x&-&7|&<&\\dfrac{-5}{-5}&&&&& \\\\ \\\\\n&&|-2x&-&7|&>&1&&&&& \\\\ \\\\ \\\\\n-2x&-&7&<&-1&\\text{or}&1&<&-2x&-&7& \\\\\n&+&7&&+7&&+7&&&+&7& \\\\\n\\hline\n&&\\dfrac{-2x}{-2}&<&\\dfrac{6}{-2}&&\\dfrac{8}{-2}&<&\\dfrac{-2x}{-2}&&& \\\\ \\\\\n&&x&>&-3&\\text{or}&-4&>&x&&&\\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } (-\\infty, -4)\\cup (-3, \\infty)\n[\/latex]\n\"(negative<\/li>\n \t
  28. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrl}\n-2&+&3|5&-&x|& \\le& 4& \\\\\n+2&&&&&&+2& \\\\\n\\hline\n&&\\dfrac{3}{3}|5&-&x|& \\le&\\dfrac{6}{3}& \\\\ \\\\\n&&|5&-&x|& \\le&2& \\\\ \\\\\n-2&\\le &5&-&x&\\le &2& \\\\\n-5&&-5&&&&-5& \\\\\n\\hline\n(-7&\\le &&-x&&\\le &-3)&(-1) \\\\\n7&\\ge &&x&&\\ge &3&\\\\ \\\\ \\end{array} \\\\ \\text{Interval notation: } [3,7]\n[\/latex]\n\"[3,7]\"<\/li>\n \t
  29. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrl}\n3&-&2|4x&-&5|&\\ge &1& \\\\\n-3&&&&&&-3& \\\\\n\\hline\n&&\\dfrac{-2}{-2}|4x&-&5|&\\ge &\\dfrac{-2}{-2}& \\\\ \\\\\n&&|4x&-&5|&\\le &1& \\\\ \\\\\n-1&\\le &4x&-&5&\\le &1& \\\\\n+5&&&+&5&&+5& \\\\\n\\hline\n\\dfrac{4}{4}&\\le &&\\dfrac{4x}{4}&&\\le &\\dfrac{6}{4}& \\\\ \\\\\n1&\\le &&x&&\\le &\\dfrac{3}{2}&\\\\ \\\\ \\end{array} \\\\ \\text{Interval notation: } [1, \\dfrac{3}{2}]\n[\/latex]\n\"[2\/2,<\/li>\n \t
  30. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrl}\n-2&-&3|-3x&-&5|&\\ge &-5& \\\\\n+2&&&&&&+2& \\\\\n\\hline\n&&\\dfrac{-3}{-3}|-3x&-&5|&\\ge &\\dfrac{-3}{-3}& \\\\ \\\\\n&&|-3x&-&5|&\\le &1& \\\\ \\\\\n-1&\\le &-3x&-&5&\\le &1& \\\\\n+5&&&+&5&&+5& \\\\\n\\hline\n\\dfrac{4}{-3}&\\le &&\\dfrac{-3x}{-3}&&\\le &\\dfrac{6}{-3}& \\\\ \\\\\n-\\dfrac{4}{3}&\\ge &&x&&\\ge &-2&\\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } [-2, -\\dfrac{4}{3}]\n[\/latex]\n\"[-6\/3,<\/li>\n \t
  31. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrl}\n-5&-&2|3x&-&6|&<&-8&&&& \\\\\n+5&&&&&&+5&&&& \\\\\n\\hline\n&&\\dfrac{-2}{-2}|3x&-&6|&<&\\dfrac{-3}{-2}&&&& \\\\ \\\\\n&&|3x&-&6|&>&\\dfrac{3}{2}&&&& \\\\ \\\\ \\\\\n3x&-&6&<&-\\dfrac{3}{2}&\\text{or}&\\dfrac{3}{2}&<&3x&-&6 \\\\ \\\\\n&+&6&&+6&&+6&&&+&6 \\\\\n\\hline\n&&\\dfrac{3x}{3}&<&\\dfrac{\\dfrac{9}{2}}{3}&&\\dfrac{\\dfrac{15}{2}}{3}&<&\\dfrac{3x}{3}&& \\\\ \\\\\n&&x&<&\\dfrac{3}{2}&\\text{or}&5&<&x&&\\\\ \\\\ \\end{array}\\\\\n\\text{Interval notation: } (-\\infty, \\dfrac{3}{2})\\cup (5, \\infty)\n[\/latex]\n\"(negative<\/li>\n \t
  32. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrrl}\n6&-&3|1&-&4x|&<&-3&&&&& \\\\\n-6&&&&&&-6&&&&& \\\\\n\\hline\n&&\\dfrac{-3}{-3}|1&-&4x|&<&\\dfrac{-9}{-3}&&&&& \\\\ \\\\\n&&|1&-&4x|&>&3&&&&& \\\\ \\\\\n1&-&4x&<&-3&\\text{or}&1&-&4x&>&3& \\\\\n-1&&&&-1&&-1&&&&-1& \\\\\n\\hline\n&&-4x&<&-4&&&&-4x&>&2& \\\\\n&&x&>&1&&\\text{or}&&x&<&-\\dfrac{1}{2}&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\infty, -\\dfrac{1}{2})\\cup (1, \\infty)\n[\/latex]\n\"(negative<\/li>\n \t
  33. [latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrcrrrl}\n4&-&4|-2x&+&6|&>&-4& \\\\\n-4&&&&&&-4& \\\\\n\\hline\n&&\\dfrac{-4}{-4}|-2x&+&6|&>&\\dfrac{-8}{-4}& \\\\ \\\\\n&&|-2x&+&6|&<&2& \\\\ \\\\\n-2&<&-2x&+&6&<&2& \\\\\n-6&&&-&6&&-6& \\\\\n\\hline\n\\dfrac{-8}{-2}&<&&\\dfrac{-2x}{-2}&&<&\\dfrac{-4}{-2}& \\\\ \\\\\n4&>&&x&&>&2&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (2,4)\n[\/latex]\n\"(2,4)\"<\/li>\n<\/ol>","rendered":"
      \n
    1. [latex]\\begin{array}{rrcrrr} -3& <& x& <& 3& \\hspace{0.25in} \\text{Interval notation: } (-3,3) \\end{array}[\/latex]\n\"(-3,3)\"<\/li>\n
    2. [latex]\\begin{array}{rrcrrr} -8& \\le& x& \\le& 8 & \\hspace{0.25in} \\text{Interval notation: } [-8,8] \\end{array}[\/latex]
      \n\"Numberline<\/li>\n
    3. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrcrrr} \\dfrac{-6}{2}&<&\\dfrac{2x}{2}&<&\\dfrac{6}{2}& \\\\ \\\\ -3&<&x&<&3& \\hspace{0.25in} \\text{Interval notation: } (-3,3) \\end{array}[\/latex]\n\"Numerline<\/li>\n
    4. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrr} -4&<&x&+&3&<&4& \\\\ -3&&&-&3&&-3& \\\\ \\hline -7&<&&x&&<&1& \\hspace{0.25in} \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-7,1)[\/latex]\n\"Numberline<\/li>\n
    5. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrr} -6&<&x&-&2&<&6& \\\\ +2&&&+&2&&+2& \\\\ \\hline -4&<&&x&&<&8& \\hspace{0.25in}\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-4,8)[\/latex]\n\"Numberline<\/li>\n
    6. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrr}-12&<&x&-&8&<&12& \\\\ +8&&&+&8&&+8& \\\\ \\hline -4&<&&x&&<&20& \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-4,20)[\/latex]\n\"Numberline<\/li>\n
    7. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrr} -3&<&x&-&7&<&3& \\\\ +7&&&+&7&&+7& \\\\ \\hline 4&<&&x&&<&10& \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (4,10)[\/latex]\n\"Numberline<\/li>\n
    8. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrr} -4&\\le &x&+&3&\\le &4& \\\\ -3&&&-&3&&-3& \\\\ \\hline -7&\\le &&x&&\\le &1& \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } [-7,1][\/latex]
      \n\"Numberline<\/li>\n
    9. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrr} -9&<&3x&-&2&<&9& \\\\ +2&&&+&2&&+2& \\\\ \\hline \\dfrac{-7}{3}&<&&\\dfrac{3x}{3}&&<&\\dfrac{11}{3}&\\\\ \\\\ -\\dfrac{7}{3}&<&&x&&<&\\dfrac{11}{3}& \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\dfrac{7}{3},\\dfrac{11}{3})[\/latex]\n\"Numberline<\/li>\n
    10. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrr} -9&<&2x&+&5&<&9& \\\\ -5&&&-&5&&-5& \\\\ \\hline \\dfrac{-14}{2}&<&&\\dfrac{2x}{2}&&<&\\dfrac{4}{2}&\\\\ \\\\ -7&<&&x&&<&2& \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-7,2)[\/latex]\n\"Numberline<\/li>\n
    11. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrr} 1&+&2|x&-&1|&\\le & 9 \\\\ -1&&&&&&-1 \\\\ \\hline &&\\dfrac{2}{2}|x&-&1|&\\le & \\dfrac{8}{2} \\\\ \\\\ &&|x&-&1|&\\le & 4 \\\\ -4&\\le &x&-&1&\\le & 4 \\\\ +1&&&+&1&\\le &+1 \\\\ \\hline -3&\\le &&x&&\\le &5& \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } [-3,5][\/latex]
      \n\"Numberline<\/li>\n
    12. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrr} 10&-&3|x&-&2|&\\ge & 4 \\\\ -10&&&&&&-10 \\\\ \\hline &&\\dfrac{-3}{-3}|x&-&2|&\\ge & \\dfrac{-6}{-3} \\\\ \\\\ &&|x&-&2|&\\le & 2 \\\\ -2&\\le &x&-&2&\\le & 2 \\\\ +2&&&+&2&&+2 \\\\ \\hline 0&\\le &&x&&\\le &4& \\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } [0,4][\/latex]
      \n\"Numberline<\/li>\n
    13. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrl} 6&-&|2x&-&5|&>&3& \\\\ -6&&&&&&-6& \\\\ \\hline &&(-|2x&-&5|&>&-3)&(-1) \\\\ &&|2x&-&5|&<&3& \\\\ -3&<&2x&-&5&<&3& \\\\ +5&&&+&5&&+5& \\\\ \\hline \\dfrac{2}{2}&<&&\\dfrac{2x}{2}&&<&\\dfrac{8}{2}& \\\\ \\\\ 1&<&&x&&<&4& \\\\ \\\\ \\end{array} \\\\ \\text{Interval notation: } (1,4)[\/latex]\n\"Numberline<\/li>\n
    14. [latex]\\begin{array}{rrcrrrrrr} x& <&-5&\\text{or}&5&<&x& \\hspace{0.25in} \\text{Interval notation: } (-\\infty,-5)\\cup (5, \\infty) \\end{array}[\/latex]\n\"Numberline<\/li>\n
    15. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrr} \\dfrac{3x}{3}&<&\\dfrac{-5}{3}&\\text{or}&\\dfrac{5}{3}&<&\\dfrac{3x}{3}& \\\\ \\\\ x&<&-\\dfrac{5}{3}&\\text{or}&\\dfrac{5}{3}&<&x&\\\\ \\\\ \\end{array} \\\\ \\text{Interval notation: } (-\\infty, -\\dfrac{5}{3})\\cup (\\dfrac{5}{3}, \\infty) \\\\[\/latex]\n\"(-<\/li>\n
    16. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrrr} x&-&4&<&-5&\\text{or}&5&<&x&-&4&\\\\ &+&4&&+4&\\text{or}&+4&&&+&4&\\\\ \\hline &&x&<&-1&\\text{or}&9&<&x&&&\\\\ \\\\ \\end{array} \\\\ \\text{Interval notation: } (-\\infty, -1)\\cup (9, \\infty)[\/latex]\n\"(-<\/li>\n
    17. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrrr} x&+&3&<&-3&\\text{or}&3&<&x&+&3& \\\\ &-&3&&-3&\\text{or}&-3&&&-&3& \\\\ \\hline &&x&<&-6&\\text{or}&0&<&x&&&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\infty, -6)\\cup (0, \\infty)\\\\[\/latex]\n\"(-<\/li>\n
    18. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrrr} 2x&-&4&<&-6&\\text{or}&6&<&2x&-&4& \\\\ &+&4&&+4&&+4&&&+&4& \\\\ \\hline &&\\dfrac{2x}{2}&<&\\dfrac{-2}{2}&&\\dfrac{10}{2}&<&\\dfrac{2x}{2}&&& \\\\ \\\\ &&x&<&-1&\\text{or}&5&<&x&&&\\\\ \\\\ \\end{array} \\\\ \\text{Interval notation: } (-\\infty, -1)\\cup (5, \\infty)\\\\[\/latex]\n\"(negative<\/li>\n
    19. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrrr} x&-&5&<&-3&\\text{or}&3&<&x&-&5& \\\\ &+&5&&+5&&+5&&&+&5& \\\\ \\hline &&x&<&2&\\text{or}&8&<&x&&&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\infty, 2)\\cup (8, \\infty)\\\\[\/latex] HAVE TO ADD 2 2 etc.\n\"(1,<\/li>\n
    20. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrl} 3&-|2&-&x|&<&1&&&&&\\\\ -3&&&&&-3&&&&&\\\\ \\hline &(-|2&-&x|&<&-2)&(-1)&&&&\\\\ &|2&-&x|&>&2&&&&&\\\\ \\\\ 2&-&x&<&-2&\\text{or}&2&<&2&-&x\\\\ -2&&&&-2&&-2&&-2&&\\\\ \\hline &&-x&<&-4&\\text{or}&0&<&-x&&\\\\ &&x&>&4&\\text{or}&0&>&x&&\\\\ \\\\ \\end{array} \\\\ \\text{Interval notation: } (-\\infty, 0)\\cup (4, \\infty)\\\\[\/latex]
      \n\"(negative<\/li>\n
    21. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrl} 4&+&3|x&-&1|&<&10&&&&\\\\ -4&&&&&&-4&&&&\\\\ \\hline &&\\dfrac{3}{3}|x&-&1|&<&\\dfrac{6}{3}&&&&\\\\ \\\\ &&|x&-&1|&<&2&&&&\\\\ \\\\ x&-&1&<&-2&\\text{or}&2&<&x&-&1\\\\ &+&1&&+1&&+1&&&+&1\\\\ \\hline &&x&<&-1&\\text{or}&3&<&x&&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\infty, -1)\\cup (3, \\infty)\\\\[\/latex]\n\"(negative<\/li>\n
    22. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrl} 3&-&2|3x&-&1|&\\ge &-7& \\\\ -3&&&&&&-3& \\\\ \\hline &&\\dfrac{-2}{-2}|3x&-&1|&\\ge &\\dfrac{-10}{-2}& \\\\ \\\\ &&|3x&-&1|&\\le &5& \\\\ \\\\ \\\\ -5&\\le &3x&-&1& \\le & 5& \\\\ +1&&&+&1&&+1& \\\\ \\hline \\dfrac{-4}{3}&\\le &&\\dfrac{3x}{3}&& \\le & \\dfrac{6}{3}& \\\\ \\\\ -\\dfrac{4}{3}&\\le &&x&& \\le & 2 &\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } [-\\dfrac{4}{3}, 2][\/latex]
      \n\"(-4\/3,<\/li>\n
    23. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrcrrrrl} 3&-&2|x&-&5|&\\le & -15&&&& \\\\ -3&&&&&&-3&&&& \\\\ \\hline &&\\dfrac{-2}{-2}|x&-&5|&\\le & \\dfrac{-18}{-2}&&&& \\\\ \\\\ &&|x&-&5|&\\ge & 9&&&& \\\\ \\\\ x&-&5&\\le &-9&\\text{or}&9&\\le &x&-&5 \\\\ &+&5&&+5&&+5&&&+&5 \\\\ \\hline &&x&\\le &-4&\\text{or}&14&\\le &x&&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\infty, -4]\\cup [14, \\infty)[\/latex]
      \n\"(-<\/li>\n
    24. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrcrrcrrrll} 4&-&6|-6&-&3x|&\\le & -5&&&& \\\\ -4&&&&&&-4&&&& \\\\ \\hline &&\\dfrac{-6}{-6}|-6&-&3x|&\\le &\\dfrac{-9}{-6}&&&& \\\\ \\\\ &&|-6&-&3x|&\\ge &\\dfrac{3}{2}&&&& \\\\ \\\\ \\\\ -6&-&3x&\\le &-\\dfrac{3}{2}&\\text{or}&\\dfrac{3}{2}&\\le &-6&-\\phantom{0}3x& \\\\ \\\\ +6&&&&+6&&+6&&+6&& \\\\ \\hline &&\\dfrac{-3x}{-3}&\\le &\\dfrac{\\dfrac{9}{2}}{-3}&&\\dfrac{\\dfrac{15}{2}}{-3}&\\le &\\dfrac{-3x}{-3}&& \\\\ \\\\ &&x&\\ge &-\\dfrac{3}{2}&\\text{or}&-\\dfrac{5}{2}&\\ge &x&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\infty, -\\dfrac{5}{2}]\\cup [-\\dfrac{3}{2}, \\infty)[\/latex]
      \n\"[-5\/2,<\/li>\n
    25. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrr} -2&-&3|4&-&2x|&\\ge & -8& \\\\ +2&&&&&&+2& \\\\ \\hline &&\\dfrac{-3}{-3}|4&-&2x|&\\ge &\\dfrac{-6}{-3}& \\\\ \\\\ &&|4&-&2x|&\\le &2& \\\\ \\\\ \\\\ -2&\\le &4&-&2x&\\le &2& \\\\ -4&&-4&&&&-4& \\\\ \\hline \\dfrac{-6}{-2}&\\le &&\\dfrac{-2x}{-2}&&\\le &\\dfrac{-2}{-2}& \\\\ \\\\ 3&\\ge &&x&&\\ge &1&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } [1,3][\/latex]
      \n\"[1,3]\"<\/li>\n
    26. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrcrr} -3&-&2|4x&-&5|&\\ge & 1& \\\\ +3&&&&&&+3& \\\\ \\hline &&\\dfrac{-2}{-2}|4x&-&5|&\\ge &\\dfrac{4}{-2}& \\\\ \\\\ &&|4x&-&5|&\\le &-2& \\\\ \\\\ &&&&&\\uparrow&& \\\\ &&&&&\\text{Cannot be true.}&&\\text{No solution.} \\end{array}[\/latex]
      \n\"Numerline<\/li>\n
    27. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrrl} 4&-&5|-2x&-&7|&<&-1&&&&& \\\\ -4&&&&&&-4&&&&& \\\\ \\hline &&\\dfrac{-5}{-5}|-2x&-&7|&<&\\dfrac{-5}{-5}&&&&& \\\\ \\\\ &&|-2x&-&7|&>&1&&&&& \\\\ \\\\ \\\\ -2x&-&7&<&-1&\\text{or}&1&<&-2x&-&7& \\\\ &+&7&&+7&&+7&&&+&7& \\\\ \\hline &&\\dfrac{-2x}{-2}&<&\\dfrac{6}{-2}&&\\dfrac{8}{-2}&<&\\dfrac{-2x}{-2}&&& \\\\ \\\\ &&x&>&-3&\\text{or}&-4&>&x&&&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\infty, -4)\\cup (-3, \\infty)[\/latex]
      \n\"(negative<\/li>\n
    28. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrl} -2&+&3|5&-&x|& \\le& 4& \\\\ +2&&&&&&+2& \\\\ \\hline &&\\dfrac{3}{3}|5&-&x|& \\le&\\dfrac{6}{3}& \\\\ \\\\ &&|5&-&x|& \\le&2& \\\\ \\\\ -2&\\le &5&-&x&\\le &2& \\\\ -5&&-5&&&&-5& \\\\ \\hline (-7&\\le &&-x&&\\le &-3)&(-1) \\\\ 7&\\ge &&x&&\\ge &3&\\\\ \\\\ \\end{array} \\\\ \\text{Interval notation: } [3,7][\/latex]
      \n\"[3,7]\"<\/li>\n
    29. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrl} 3&-&2|4x&-&5|&\\ge &1& \\\\ -3&&&&&&-3& \\\\ \\hline &&\\dfrac{-2}{-2}|4x&-&5|&\\ge &\\dfrac{-2}{-2}& \\\\ \\\\ &&|4x&-&5|&\\le &1& \\\\ \\\\ -1&\\le &4x&-&5&\\le &1& \\\\ +5&&&+&5&&+5& \\\\ \\hline \\dfrac{4}{4}&\\le &&\\dfrac{4x}{4}&&\\le &\\dfrac{6}{4}& \\\\ \\\\ 1&\\le &&x&&\\le &\\dfrac{3}{2}&\\\\ \\\\ \\end{array} \\\\ \\text{Interval notation: } [1, \\dfrac{3}{2}][\/latex]
      \n\"[2\/2,<\/li>\n
    30. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrl} -2&-&3|-3x&-&5|&\\ge &-5& \\\\ +2&&&&&&+2& \\\\ \\hline &&\\dfrac{-3}{-3}|-3x&-&5|&\\ge &\\dfrac{-3}{-3}& \\\\ \\\\ &&|-3x&-&5|&\\le &1& \\\\ \\\\ -1&\\le &-3x&-&5&\\le &1& \\\\ +5&&&+&5&&+5& \\\\ \\hline \\dfrac{4}{-3}&\\le &&\\dfrac{-3x}{-3}&&\\le &\\dfrac{6}{-3}& \\\\ \\\\ -\\dfrac{4}{3}&\\ge &&x&&\\ge &-2&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } [-2, -\\dfrac{4}{3}][\/latex]
      \n\"[-6\/3,<\/li>\n
    31. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrl} -5&-&2|3x&-&6|&<&-8&&&& \\\\ +5&&&&&&+5&&&& \\\\ \\hline &&\\dfrac{-2}{-2}|3x&-&6|&<&\\dfrac{-3}{-2}&&&& \\\\ \\\\ &&|3x&-&6|&>&\\dfrac{3}{2}&&&& \\\\ \\\\ \\\\ 3x&-&6&<&-\\dfrac{3}{2}&\\text{or}&\\dfrac{3}{2}&<&3x&-&6 \\\\ \\\\ &+&6&&+6&&+6&&&+&6 \\\\ \\hline &&\\dfrac{3x}{3}&<&\\dfrac{\\dfrac{9}{2}}{3}&&\\dfrac{\\dfrac{15}{2}}{3}&<&\\dfrac{3x}{3}&& \\\\ \\\\ &&x&<&\\dfrac{3}{2}&\\text{or}&5&<&x&&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\infty, \\dfrac{3}{2})\\cup (5, \\infty)[\/latex]\n\"(negative<\/li>\n
    32. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrrl} 6&-&3|1&-&4x|&<&-3&&&&& \\\\ -6&&&&&&-6&&&&& \\\\ \\hline &&\\dfrac{-3}{-3}|1&-&4x|&<&\\dfrac{-9}{-3}&&&&& \\\\ \\\\ &&|1&-&4x|&>&3&&&&& \\\\ \\\\ 1&-&4x&<&-3&\\text{or}&1&-&4x&>&3& \\\\ -1&&&&-1&&-1&&&&-1& \\\\ \\hline &&-4x&<&-4&&&&-4x&>&2& \\\\ &&x&>&1&&\\text{or}&&x&<&-\\dfrac{1}{2}&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (-\\infty, -\\dfrac{1}{2})\\cup (1, \\infty)[\/latex]\n\"(negative<\/li>\n
    33. [latex]\\phantom{a}[\/latex]
      \n[latex]\\begin{array}[t]{rrrcrrrl} 4&-&4|-2x&+&6|&>&-4& \\\\ -4&&&&&&-4& \\\\ \\hline &&\\dfrac{-4}{-4}|-2x&+&6|&>&\\dfrac{-8}{-4}& \\\\ \\\\ &&|-2x&+&6|&<&2& \\\\ \\\\ -2&<&-2x&+&6&<&2& \\\\ -6&&&-&6&&-6& \\\\ \\hline \\dfrac{-8}{-2}&<&&\\dfrac{-2x}{-2}&&<&\\dfrac{-4}{-2}& \\\\ \\\\ 4&>&&x&&>&2&\\\\ \\\\ \\end{array}\\\\ \\text{Interval notation: } (2,4)[\/latex]
      \n\"(2,4)\"<\/li>\n<\/ol>\n","protected":false},"author":90,"menu_order":32,"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-1763","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\/1763","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\/1763\/revisions"}],"predecessor-version":[{"id":1764,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1763\/revisions\/1764"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1763\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=1763"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter-type?post=1763"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=1763"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=1763"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}