"

Answer Key 2.5

  1. [latex]x=\pm 8[/latex]
  2. [latex]n=\pm 7[/latex]
  3. [latex]b=\pm 1[/latex]
  4. [latex]x=\pm 2[/latex]
  5. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrr} 5&+&8a&=&53 \\ -5&&&&-5 \\ \hline &&\dfrac{8a}{8}&=&\dfrac{48}{8} \\ \\ &&a&=&6 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrl} 5&+&8a&=&-53 \\ -5&&&&-5 \\ \hline &&\dfrac{8a}{8}&=&\dfrac{-58}{8} \\ \\ &&a&=&-\dfrac{58}{8}\text{ or }-7\dfrac{1}{4} \end{array} \end{array}[/latex]
  6. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrl} 9n&+&8&=&46 \\ &-&8&&-8 \\ \hline &&\dfrac{9n}{9}&=&\dfrac{38}{9} \\ \\ &&n&=&\dfrac{38}{9}\text{ or }4\dfrac{2}{9} \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 9n&+&8&=&-46 \\ &-&8&&-8 \\ \hline &&\dfrac{9n}{9}&=&\dfrac{-54}{9} \\ \\ &&n&=&-6 \end{array} \end{array}[/latex]
  7. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrr} 3k&+&8&=&2 \\ &-&8&&-8 \\ \hline &&\dfrac{3k}{3}&=&\dfrac{-6}{3} \\ \\ &&k&=&-2 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 3k&+&8&=&-2 \\ &-&8&&-8 \\ \hline &&\dfrac{3k}{3}&=&\dfrac{-10}{3} \\ \\ &&k&=&-\dfrac{10}{3} \end{array} \end{array}[/latex]
  8. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrl} 3&-&x&=&\phantom{-}6 \\ -3&&&&-3 \\ \hline &&(-x&=&\phantom{-}3)(-1) \\ &&x&=&-3 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrl} 3&-&x&=&-6 \\ -3&&&&-3 \\ \hline &&(-x&=&-9)(-1) \\ &&x&=&\phantom{-}9 \end{array} \end{array}[/latex]
  9. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrl} \dfrac{-7}{-7}\left| -3-3r \right|&=&\dfrac{-21}{-7} \\ |-3-3r|&=&3 \end{array}[/latex]
    [latex]\phantom{1}[/latex]
    [latex]\begin{array}{ll} \begin{array}{rrrrr} -3&-&3r&=&3 \\ +3&&&&+3 \\ \hline &&\dfrac{-3r}{-3}&=&\dfrac{6}{-3} \\ \\ &&r&=&-2 \end{array} & \hspace{0.5in} \begin{array}{rrrrr} -3&-&3r&=&-3 \\ +3&&&&+3 \\ \hline &&\dfrac{-3r}{-3}&=&\dfrac{0}{-3} \\ \\ &&r&=&0 \end{array} \end{array}[/latex]
  10. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrr} |2+2b|&+&1&=&3 \\ &-&1&&-1 \\ \hline |2+2b|&&&=&2 \\ \end{array}[/latex]
    [latex]\begin{array}[t]{ll} \\ \begin{array}[t]{rrrrr} 2&+&2b&=&2 \\ -2&&&&-2 \\ \hline &&2b&=&0 \\ &&b&=&0 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 2&+&2b&=&-2 \\ -2&&&&-2 \\ \hline &&\dfrac{2b}{2}&=&\dfrac{-4}{2} \\ \\ &&b&=&-2 \end{array} \end{array}[/latex]
  11. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrl} \dfrac{7}{7}|-7x-3|&=&\dfrac{21}{7} \\ |-7x-3|&=&3 \end{array}[/latex]
    [latex]\begin{array}[t]{ll}\\ \begin{array}[t]{rrrrr} -7x&-&3&=&3 \\ &+&3&&+3 \\ \hline &&\dfrac{-7x}{-7}&=&\dfrac{6}{-7} \\ \\ &&x&=&-\dfrac{6}{7} \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} -7x&-&3&=&-3 \\ &+&3&&+3 \\ \hline &&-7x&=&0 \\ &&x&=&0 \end{array} \end{array}[/latex]
  12. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrr} -4&-&3n&=&2 \\ +4&&&&+4 \\ \hline &&\dfrac{-3n}{-3}&=&\dfrac{6}{-3} \\ \\ &&n&=&-2 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} -4&-&3n&=&-2 \\ +4&&&&+4 \\ \hline &&\dfrac{-3n}{-3}&=&\dfrac{2}{-3} \\ \\ &&n&=&-\dfrac{2}{3} \end{array} \end{array}[/latex]
  13. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrrr} 8|5p &+&8|&-&5&=&11 \\ &&&+&5&&+5 \\ \hline &&\dfrac{8}{8}|5p &+&8|&=&\dfrac{16}{8} \\ &&|5p &+&8|&=&2 \end{array}[/latex]
    [latex]\begin{array}[t]{ll}\\ \begin{array}[t]{rrrrr} 5p&+&8&=&2 \\ &-&8&&-8 \\ \hline &&\dfrac{5p}{5}&=&\dfrac{-6}{5} \\ \\ &&p&=&-\dfrac{6}{5} \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 5p&+&8&=&-2 \\ &-&8&&-8 \\ \hline &&\dfrac{5p}{5}&=&\dfrac{-10}{5} \\ \\ &&p&=&-2 \end{array} \end{array}[/latex]
  14. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrrl} 3&-&|6n&+&7|&=&-40 \\ -3&&&&&&-3 \\ \hline &&(-|6n&+&7|&=&-43)(-1) \\ &&|6n&+&7|&=&43 \end{array}[/latex]
    [latex]\begin{array}[t]{ll}\\ \begin{array}[t]{rrrrr} 6n&+&7&=&43 \\ &-&7&&-7 \\ \hline &&\dfrac{6n}{6}&=&\dfrac{36}{6} \\ \\ &&n&=&6 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 6n&+&7&=&-43 \\ &-&7&&-7 \\ \hline &&\dfrac{6n}{6}&=&\dfrac{-50}{6} \\ \\ &&n&=&-\dfrac{25}{3} \end{array} \end{array}[/latex]
  15. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrrr} 5|3&+&7m|&+&1&=&51 \\ &&&-&1&&-1 \\ \hline &&\dfrac{5}{5}|3&+&7m|&=&\dfrac{50}{5} \\ &&|3&+&7m|&=&10 \end{array}[/latex]
    [latex]\begin{array}[t]{ll}\\ \begin{array}[t]{rrrrr} 3&+&7m&=&10 \\ -3&&&&-3 \\ \hline &&\dfrac{7m}{7}&=&\dfrac{7}{7} \\ \\ &&m&=&1 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 3&+&7m&=&-10 \\ -3&&&&-3 \\ \hline &&\dfrac{7m}{7}&=&\dfrac{-13}{7} \\ \\ &&m&=&-\dfrac{13}{7} \end{array} \end{array}[/latex]
  16. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrrr} 4|r&+&7|&+&3&=&59 \\ &&&-&3&&-3 \\ \hline &&\dfrac{4}{4}|r&+&7|&=&\dfrac{56}{4} \\ &&|r&+&7|&=&14 \end{array}[/latex]
    [latex]\begin{array}[t]{ll}\\ \begin{array}[t]{rrrrr} r&+&7&=&14 \\ &-&7&&-7 \\ \hline &&r&=&7 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} r&+&7&=&-14 \\ &-&7&&-7 \\ \hline &&r&=&-21 \end{array} \end{array}[/latex]
  17. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrrr} -7&+&8|-7x&-&3|&=&73 \\ +7&&&&&&+7 \\ \hline &&\dfrac{8}{8}|-7x&-&3|&=&\dfrac{80}{8} \\ &&|-7x&-&3|&=&10 \end{array}[/latex]
    [latex]\phantom{1}[/latex]
    [latex]\begin{array}{ll} \begin{array}{rrrrr} -7x&-&3&=&10 \\ &+&3&&+3 \\ \hline &&\dfrac{-7x}{-7}&=&\dfrac{13}{-7} \\ \\ &&x&=&-\dfrac{13}{7} \end{array} & \hspace{0.5in} \begin{array}{rrrrr} -7x&-&3&=&-10 \\ &+&3&&+3 \\ \hline &&\dfrac{-7x}{-7}&=&\dfrac{-7}{-7} \\ \\ &&x&=&1 \end{array} \end{array}[/latex]
  18. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{rrrrrrr} 8|3&-&3n|&-&5&=&91 \\ &&&+&5&&+5 \\ \hline &&\dfrac{8}{8}|3&-&3n|&=&\dfrac{96}{8} \\ &&|3&-&3n|&=&12 \end{array}[/latex]
    [latex]\begin{array}[t]{ll}\\ \begin{array}[t]{rrrrr} 3&-&3n&=&12 \\ -3&&&&-3 \\ \hline &&\dfrac{-3n}{-3}&=&\dfrac{9}{-3} \\ \\ &&n&=&-3 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 3&-&3n&=&-12 \\ -3&&&&-3 \\ \hline &&\dfrac{-3n}{-3}&=&\dfrac{-15}{-3} \\ \\ &&n&=&5 \end{array} \end{array}[/latex]
  19. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} 5x&+&3&=&2x&-&1 \\ -2x&-&3&&-2x&-&3 \\ \hline &&\dfrac{3x}{3}&=&\dfrac{-4}{3}&& \\ \\ &&x&=&-\dfrac{4}{3}&& \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrrrr} 5x&+&3&=&-2x&+&1 \\ +2x&-&3&&+2x&-&3 \\ \hline &&\dfrac{7x}{7}&=&\dfrac{-2}{7}&& \\ \\ &&x&=&-\dfrac{2}{7}&& \end{array} \end{array}[/latex]
  20. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} 2&+&3x&=&4&-&2x \\ -2&+&2x&&-2&+&2x \\ \hline &&\dfrac{5x}{5}&=&\dfrac{2}{5}&& \\ \\ &&x&=&\dfrac{2}{5}&& \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrrrr} 2&+&3x&=&-4&+&2x \\ -2&-&2x&&-2&-&2x \\ \hline &&x&=&-6&& \end{array} \end{array}[/latex]
  21. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} 3x&-&4&=&2x&+&3 \\ -2x&+&4&&-2x&+&4 \\ \hline &&x&=&7&& \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrlrr} 3x&-&4&=&-2x&-&3 \\ +2x&+&4&&+2x&+&4 \\ \hline &&\dfrac{5x}{5}&=&\dfrac{1}{5}&& \\ \\ &&x&=&\dfrac{1}{5}&& \end{array} \end{array}[/latex]
  22. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} 2x&-&5&=&3x&+&4 \\ -3x&+&5&&-3x&+&5 \\ \hline &&-x&=&9&& \\ &&x&=&-9&& \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrlrr} 2x&-&5&=&-3x&-&4 \\ +3x&+&5&&+3x&+&5 \\ \hline &&\dfrac{5x}{5}&=&\dfrac{1}{5}&& \\ \\ &&x&=&\dfrac{1}{5}&& \end{array} \end{array}[/latex]
  23. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} 4x&-&2&=&6x&+&3 \\ -6x&+&2&&-6x&+&2 \\ \hline &&\dfrac{-2x}{-2}&=&\dfrac{5}{-2}&& \\ \\ &&x&=&-\dfrac{5}{2}&& \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrrrr} 4x&-&2&=&-6x&-&3 \\ +6x&+&2&&+6x&+&2 \\ \hline &&\dfrac{10x}{10}&=&\dfrac{-1}{10}&& \\ \\ &&x&=&-\dfrac{1}{10}&& \end{array} \end{array}[/latex]
  24. [latex]\phantom{a}[/latex]
    [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} 3x&+&2&=&2x&-&3 \\ -2x&-&2&&-2x&-&2 \\ \hline &&x&=&-5&& \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrlrr} 3x&+&2&=&-2x&+&3 \\ +2x&-&2&&+2x&-&2 \\ \hline &&\dfrac{5x}{5}&=&\dfrac{1}{5}&& \\ \\ &&x&=&\dfrac{1}{5}&& \end{array} \end{array}[/latex]

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

Share This Book