Fractionmia

Problem:



Solution:

$\begin{eqnarray*} & & \frac{\frac{1}{x} + \frac{1}{\frac{1}{x} + 1}}{\frac{1}{x} - \frac{1}{\frac{1}{x} - 1}} \\ &=& \frac{\frac{1}{x} + \frac{x}{1 + x}}{\frac{1}{x} - \frac{x}{1 - x}} \\ &=& \frac{\frac{1 + x + x^2}{x(1 + x)}}{\frac{1 - x - x^2}{x(1 - x)}} \\ &=& \frac{\frac{1 + x + x^2}{1 + x}}{\frac{1 - x - x^2}{1 - x}} \\ &=& \frac{(1 + x + x^2)(1 - x)}{(1 - x - x^2)(1 + x)} \end{eqnarray*}$