When are both fractions integers? The sum of absolute values of all real numbers $x$, such that both of the fractions $\displaystyle \frac{x^2+4x−17}{x^2−6x−5}$ and $\displaystyle \frac{1−x}{1+x}$ are integers, can be written as $\displaystyle \frac{a}{b}$, where $a$ and $b$ are coprime positive integers. What is the value of $a+b$?

Note that 
$$\frac{1+x}{1-x}=\frac{2}{1+x}-1$$
and
$$\frac{x^2+4x-17}{x^2-6x-5} = \frac{10x-12}{x^2-6x-5}+1.$$
We need both of these to be integers. So $\frac{2}{1+x}$ must be an integer, and happens when, for example, $x=-3,-2,0,1$ (you also need to check rational values). You can plug these into the second fraction to test its integrality. Then take the sum of the appropriate values. 
