Answer Key 8.6

[latexpage]

1. \(\begin{array}{rrrrr}
   \\ \\ \\ \\ \\ \\ \\ \\
   \text{LCD}&=&5(2)&& \\ \\
   2(m&-&1)&=&5(8) \\
   2m&-&2&=&40 \\
   &+&2&&+2 \\
   \midrule
   &&\dfrac{2m}{2}&=&\dfrac{42}{2} \\ \\
   &&m&=&21
   \end{array}\)
2. \(\begin{array}{rrrrl}
   \\ \\ \\ \\ \\ \\ \\ \\
   \text{LCD}&=&2(x&-&8) \\ \\
   8(x&-&8)&=&\phantom{+}2(8) \\
   8x&-&64&=&\phantom{+}16 \\
   &+&64&&+64 \\
   \midrule
   &&\dfrac{8x}{8}&=&\phantom{+}\dfrac{80}{8} \\ \\
   &&x&=&\phantom{+}10
   \end{array}\)
3. \(\begin{array}{rrrrl}
   \\ \\ \\ \\ \\ \\ \\ \\
   \text{LCD}&=&9(p&-&4) \\ \\
   2(p&-&4)&=&9(10) \\
   2p&-&8&=&90 \\
   &+&8&&+8 \\
   \midrule
   &&\dfrac{2p}{2}&=&\dfrac{98}{2} \\ \\
   &&p&=&49
   \end{array}\)
4. \(\begin{array}{rllll}
   \\ \\ \\ \\ \\ \\ \\ \\
   \text{LCD}&=&9(n&+&2) \\ \\
   9(9)&=&3(n&+&2) \\
   81&=&3n&+&6 \\
   -6&&&-&6 \\
   \midrule
   \dfrac{75}{3}&=&\dfrac{3n}{3}&& \\ \\
   n&=&25&&
   \end{array}\)
5. \(\begin{array}{rrlrl}
   \\ \\ \\ \\ \\ \\ \\ \\ \\
   \text{LCD}&=&10(a&+&2) \\ \\
   3(a&+&2)&=&10(a) \\
   3a&+&6&=&10a \\
   -3a&&&&-3a \\
   \midrule
   &&\dfrac{6}{7}&=&\dfrac{7a}{7} \\ \\
   &&a&=&\dfrac{6}{7}
   \end{array}\)
6. \(\begin{array}{rrrrrrr}
   \\ \\ \\ \\ \\
   \text{LCD}&=&3(4)&&&& \\ \\
   4(x&+&1)&=&3(x&+&3) \\
   4x&+&4&=&3x&+&9 \\
   -3x&-&4&&-3x&-&4 \\
   \midrule
   &&x&=&5&&
   \end{array}\)
7. \(\begin{array}{rrrrcrr}
   \\ \\ \\ \\ \\ \\ \\ \\
   \text{LCD}&=&3(p&+&4)&& \\ \\
   2(3)&=&(p&+&4)(p&+&5) \\
   6&=&p^2&+&9p&+&20 \\
   -6&&&&&-&6 \\
   \midrule
   0&=&p^2&+&9p&+&14 \\
   0&=&(p&+&7)(p&+&2) \\ \\
   p&=&-2,&-7&&& \\
   \end{array}\)
8. \(\begin{array}{rrrrcrr}
   \\ \\ \\ \\ \\ \\ \\ \\
   \text{LCD}&=&10(n&+&1)&& \\ \\
   5(10)&=&(n&-&4)(n&+&1) \\
   50&=&n^2&-&3n&-&4 \\
   -50&&&&&-&50 \\
   \midrule
   0&=&n^2&-&3n&-&54 \\
   0&=&(n&-&9)(n&+&6) \\ \\
   n&=&9,&-6&&&
   \end{array}\)
9. \(\begin{array}{rrcrrrl}
   \\ \\ \\ \\ \\ \\ \\ \\
   \text{LCD}&=&5(x&-&2)&& \\ \\
   (x&+&5)(x&-&2)&=&5(6) \\
   x^2&+&3x&-&10&=&\phantom{-}30 \\
   &&&-&30&&-30 \\
   \midrule
   x^2&+&3x&-&40&=&0 \\
   (x&-&5)(x&+&8)&=&0 \\ \\
   &&&&x&=&5, -7
   \end{array}\)
10. \(\begin{array}{rrrrcrr}
    \\ \\ \\ \\ \\ \\ \\ \\
    \text{LCD}&=&5(x&-&3)&& \\ \\
    20&=&(x&-&3)(x&+&5) \\
    20&=&x^2&+&2x&-&15 \\
    -20&&&&&-&20 \\
    \midrule
    0&=&x^2&+&2x&-&35 \\
    0&=&(x&-&5)(x&+&7) \\ \\
    x&=&5,&-7&&&
    \end{array}\)
11. \(\begin{array}{rrcrrrl}
    \\ \\ \\ \\ \\ \\ \\ \\
    \text{LCD}&=&4(m&-&4)&& \\ \\
    (m&+&3)(m&-&4)&=&4(11) \\
    m^2&-&m&-&12&=&\phantom{-}44 \\
    &&&-&44&&-44 \\
    \midrule
    (m^2&-&m&-&56)&=&0 \\
    (m&-&8)(m&+&7)&=&0 \\ \\
    &&&&m&=&8, -7
    \end{array}\)
12. \(\begin{array}{rrcrrrl}
    \\ \\ \\ \\ \\ \\ \\ \\
    \text{LCD}&=&8(x&-&1)&& \\ \\
    (x&-&5)(x&-&1)&=&4(8) \\
    x^2&-&6x&+&5&=&\phantom{-}32 \\
    &&&-&32&&-32 \\
    \midrule
    x^2&-&6x&-&27&=&0 \\
    (x&-&9)(x&+&3)&=&0 \\ \\
    &&&&x&=&9, -3
    \end{array}\)