Find the sum of all $x$ such that $y = 2 - \frac{9}{x+1}$ has an integer solution $(x, y)$

Question

Find the sum of all $x$ such that $y = 2 - \frac{9}{x + 1}$ has an integer solution $(x, y)$

I made $x$ the subject of the formula, i.e., wrote $$x = \left(\frac{9}{2 - y}\right) - 1 .$$ I don't know what to do after that. I need the full solution. Thanks.

Answer 1

Hint Rearranging gives $\frac{9}{x} = 1 - y$. In particular, for any solution $(x, y)$, $\frac{9}{x}$ is an integer, so (1) if $x$ is a solution, so is $-x$, and (2) there are only finitely many solutions.

Answer 2

Hint: Show that the equation is equivalent to $$ (2-y)(x+1)=9 $$