Answer Key 8.4

  1. \dfrac{2+4}{a+3}=\dfrac{6}{a+3}
  2. \dfrac{x^2-(6x-8)}{x-2}\Rightarrow \dfrac{x^2-6x+8}{x-2}\Rightarrow \dfrac{(x-4)\cancel{(x-2)}}{\cancel{(x-2)}}\Rightarrow x-4
  3. \dfrac{t^2+4t+2t-7}{t-1}\Rightarrow \dfrac{t^2+6t-7}{t-1}\Rightarrow \dfrac{(t+7)\cancel{(t-1)}}{\cancel{(t-1)}}\Rightarrow t+7
  4. \dfrac{a^2+3a-4}{a^2+5a-6}\Rightarrow \dfrac{(a+4)\cancel{(a-1)}}{(a+6)\cancel{(a-1)}}\Rightarrow \dfrac{a+4}{a+6}
  5. \text{LCD}=24r\hspace{0.25in} \dfrac{5}{6r}\cdot \dfrac{4}{4}-\dfrac{5}{8r}\cdot \dfrac{3}{3}\Rightarrow \dfrac{20}{24r}-\dfrac{15}{24r}\Rightarrow \dfrac{5}{24r}
  6. \text{LCD}=x^2y^2\hspace{0.25in} \dfrac{7}{xy^2}\cdot \dfrac{x}{x}+\dfrac{3}{x^2y}\cdot \dfrac{y}{y}\Rightarrow \dfrac{7x+3y}{x^2y^2}
  7. \text{LCD}=18t^3\hspace{0.25in} \dfrac{8}{9t^3}\cdot \dfrac{2}{2}+\dfrac{5}{6t^2}\cdot \dfrac{3t}{3t}\Rightarrow \dfrac{15t+16}{18t^3}
  8. \text{LCD}=24\hspace{0.25in} \dfrac{(x+5)(3)}{(8)(3)}+\dfrac{(x-3)(2)}{(12)(2)}\Rightarrow \dfrac{3x+15+2x-6}{24}\Rightarrow \dfrac{5x+9}{24}
  9. \text{LCD}=4x \hspace{0.25in} \dfrac{x-1}{4x}-\dfrac{4(2x+3)}{4\cdot x}\Rightarrow \dfrac{x-1-8x-12}{4x}\Rightarrow \dfrac{-7x-13}{4x}
  10. \text{LCD}=c^2d^2 \hspace{0.25in} \dfrac{(2c-d)(d)}{c^2d(d)}-\dfrac{(c+d)(c)}{cd^2(c)}\Rightarrow \dfrac{2cd-d^2-c^2-cd}{c^2d^2}\Rightarrow \dfrac{cd-c^2-d^2}{c^2d^2}
  11. \text{LCD}=2x^2y^2 \hspace{0.25in} \dfrac{(5x+3y)(y)}{(2x^2y)(y)}-\dfrac{(3x+4y)(2x)}{(xy^2)(2x)}\Rightarrow \dfrac{5xy+3y^2-6x^2-8xy}{2x^2y^2}\Rightarrow
    \dfrac{3y^2-3xy-6x^2}{2x^2y^2}
  12. \text{LCD} = (x - 1)(x + 1)\hspace{0.25in} \dfrac{2(x+1)}{(x-1)(x+1)}+\dfrac{2(x-1)}{(x+1)(x-1)}\Rightarrow \dfrac{2x+2+2x-2}{(x+1)(x-1)}\Rightarrow
    \dfrac{4x}{(x+1)(x-1)}
  13. \text{LCD}=(x+3)(x+2)(x+1)\hspace{0.25in} \dfrac{x(x+1)}{(x+3)(x+2)(x+1)}-\dfrac{2(x+3)}{(x+3)(x+2)(x+1)}\Rightarrow \\
    \dfrac{x^2+x-2x-6}{(x+3)(x+2)(x+1)}\Rightarrow \dfrac{x^2-x-6}{(x+3)(x+2)(x+1)}\Rightarrow \dfrac{(x-3)\cancel{(x+2)}}{(x+3)\cancel{(x+2)}(x+1)}\Rightarrow \\
    \dfrac{x-3}{(x+3)(x+1)}
  14. \text{LCD}=(x-1)(x+1)(x+4) \hspace{0.25in} \dfrac{2x(x+4)}{(x-1)(x+1)(x+4)}-\dfrac{3(x-1)}{(x-1)(x+1)(x+4)}\Rightarrow \\
    \dfrac{2x^2+8x-3x+3}{(x-1)(x+1)(x+4)}\Rightarrow \dfrac{2x^2+5x+3}{(x-1)(x+1)(x+4)}\Rightarrow \dfrac{(2x+3)\cancel{(x+1)}}{(x-1)\cancel{(x+1)}(x+4)}\Rightarrow \\
    \dfrac{2x+3}{(x-1)(x+4)}
  15. \text{LCD}=(x+7)(x+8)(x+6) \hspace{0.25in} \dfrac{x(x+6)}{(x+7)(x+8)(x+6)}-\dfrac{7(x+8)}{(x+7)(x+8)(x+6)}\Rightarrow \\
    \dfrac{x^2+6x-7x-56}{(x+7)(x+8)(x+6)}\Rightarrow \dfrac{x^2-x-56}{(x+7)(x+8)(x+6)}\Rightarrow \dfrac{(x-8)\cancel{(x+7)}}{\cancel{(x+7)}(x+8)(x+6)}\Rightarrow \\
    \dfrac{x-8}{(x+8)(x+6)}
  16. \text{LCD}=(x-3)(x+3)(x-2) \hspace{0.25in} \dfrac{2x(x-2)}{(x-3)(x+3)(x-2)}+\dfrac{5(x-3)}{(x-3)(x+3)(x-2)}\Rightarrow \\
    \dfrac{2x^2-4x+5x-15}{(x-3)(x+3)(x-2)}\Rightarrow \dfrac{2x^2+x-15}{(x-3)(x+3)(x-2)}\Rightarrow \dfrac{\cancel{(x+3)}(2x-5)}{(x-3)\cancel{(x+3)}(x-2)}\Rightarrow \\
    \dfrac{2x-5}{(x-3)(x-2)}
  17. \text{LCD}=(x-3)(x+2)(x+3) \hspace{0.25in} \dfrac{5x(x+3)}{(x-3)(x+2)(x+3)}-\dfrac{18(x+2)}{(x-3)(x+2)(x+3)}\Rightarrow \\
    \dfrac{5x^2+15x-18x-36}{(x-3)(x+2)(x+3)}\Rightarrow \dfrac{5x^2-3x-36}{(x-3)(x+2)(x+3)}\Rightarrow \dfrac{\cancel{(x-3)}(5x+12)}{\cancel{(x-3)}(x+2)(x+3)}\Rightarrow \\
    \dfrac{5x+12}{(x+2)(x+3)}
  18. \text{LCD}=(x-3)(x+1)(x-2) \hspace{0.25in} \dfrac{4x(x-2)}{(x-3)(x+1)(x-2)}-\dfrac{3(x+1)}{(x-3)(x+1)(x-2)}\Rightarrow \\
    \dfrac{4x^2-8x-3x-3}{(x-3)(x+1)(x-2)}\Rightarrow \dfrac{4x^2-11x-3}{(x-3)(x+1)(x-2)}\Rightarrow \dfrac{(4x+1)\cancel{(x-3)}}{\cancel{(x-3)}(x+1)(x-2)}\Rightarrow \\
    \dfrac{4x+1}{(x+1)(x-2)}

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