Answer Key 8.5

  1. \dfrac{\left(1+\dfrac{1}{x}\right)x^2}{\left(1-\dfrac{1}{x^2}\right)x^2}\Rightarrow \dfrac{x^2+x}{x^2-1}\Rightarrow \dfrac{x\cancel{(x+1)}}{\cancel{(x+1)}(x-1)}\Rightarrow \dfrac{x}{x-1}
  2. \dfrac{\left(1-\dfrac{1}{y^2}\right)y^2}{\left(1+\dfrac{1}{y}\right)y^2}\Rightarrow \dfrac{y^2-1}{y^2+y}\Rightarrow \dfrac{(y-1)\cancel{(y+1)}}{y\cancel{(y+1)}}\Rightarrow \dfrac{y-1}{y}
  3. \dfrac{\left(\dfrac{a}{b}+2\right)b^2}{\left(\dfrac{a^2}{b^2}-4\right)b^2}\Rightarrow \dfrac{ab+2b^2}{a^2-4b^2}\Rightarrow \dfrac{b\cancel{(a+2b)}}{\cancel{(a+2b)}(a-2b)}\Rightarrow \dfrac{b}{a-2b}
  4. \dfrac{\left(\dfrac{1}{y^2}-9\right)y^2}{\left(\dfrac{1}{y}+3\right)y^2}\Rightarrow \dfrac{1-9y^2}{y+3y^2}\Rightarrow \dfrac{(1-3y)\cancel{(1+3y)}}{y\cancel{(1+3y)}}\Rightarrow \dfrac{1-3y}{y}
  5. \dfrac{\left(\dfrac{1}{a^2}-\dfrac{1}{a}\right)a^2}{\left(\dfrac{1}{a^2}+\dfrac{1}{a}\right)a^2}\Rightarrow \dfrac{1-a}{1+a}
  6. \dfrac{\left(\dfrac{1}{b}+\dfrac{1}{2}\right)2b(b^2-1)}{\left(\dfrac{4}{b^2-1}\right)2b(b^2-1)}\Rightarrow \dfrac{2b(b^2-1)+b(b^2-1)}{4(2b)}\Rightarrow \dfrac{2b^2-2+b^3-b}{8b}\Rightarrow \\ \\
    \dfrac{b^3+2b^2-b-2}{8b}\Rightarrow \dfrac{(b-1)(b+1)(b+2)}{8b}
  7. \dfrac{\left(x+2-\dfrac{9}{x+2}\right)(x+2)}{\left(x+1+\dfrac{x-7}{x+2}\right)(x+2)}\Rightarrow \dfrac{(x+2)(x+2)-9}{(x+1)(x+2)+x-7}\Rightarrow \dfrac{x^2+4x+4-9}{x^2+3x+2+x-7}\Rightarrow \\
    \dfrac{x^2+4x-5}{x^2+4x-5}\Rightarrow 1
  8. \dfrac{\left(a-3+\dfrac{a-3}{a+2}\right)(a+2)}{\left(a+4-\dfrac{4a+5}{a+2}\right)(a+2)}\Rightarrow \dfrac{(a-3)(a+2)+a-3}{(a+4)(a+2)-4a+5}\Rightarrow \dfrac{a^2-a-6+a-3}{a^2+6a+8-4a+5}\Rightarrow \\
    \dfrac{a^2-9}{a^2+2a+13}\Rightarrow \dfrac{(a-3)(a+3)}{a^2+2a+13}
  9. \dfrac{\left(\dfrac{x+y}{y}+\dfrac{y}{x-y}\right)y(x-y)}{\left(\dfrac{y}{x-y}\right)y(x-y)}\Rightarrow \dfrac{(x+y)(x-y)+y(y)}{y(y)}\Rightarrow \\ \\
    \dfrac{x^2-y^2+y^2}{y^2}  \Rightarrow \dfrac{x^2}{y^2}
  10. \dfrac{\left(\dfrac{a-b}{a}-\dfrac{a}{a+b}\right)a(a+b)}{\left(\dfrac{b^2}{a+b}\right)a(a+b)}\Rightarrow \dfrac{(a-b)(a+b)-a(a)}{b^2(a)}\Rightarrow \\ \\
    \dfrac{a^2-b^2-a^2}{ab^2}\Rightarrow \dfrac{-b^2}{ab^2}
  11. \dfrac{\left(\dfrac{x-y}{y}+\dfrac{x+y}{x-y}\right)y(x-y)}{\left(\dfrac{y}{x-y}\right)y(x-y)}\Rightarrow \dfrac{(x-y)(x-y)+(x+y)(y)}{y(y)}\Rightarrow \\
    \dfrac{x^2-2xy+y^2+xy+y^2}{y^2}\Rightarrow \dfrac{x^2-xy+2y^2}{y^2}
  12. \dfrac{\left(\dfrac{x-2}{x+2}-\dfrac{x+2}{x-2}\right)(x+2)(x-2)}{\left(\dfrac{x-2}{x+2}+\dfrac{x+2}{x-2}\right)(x+2)(x-2)}\Rightarrow \dfrac{x^2-4x+4-(x^2+4x+4)}{x^2-4x+4+x^2+4x+4}\Rightarrow \dfrac{-8x}{2x^2+8}\Rightarrow \\
    \dfrac{\cancel{2}(-4x)}{\cancel{2}(x^2+4)}\Rightarrow \dfrac{-4x}{x^2+4}

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