I have tried to attack this problem in numerous ways, but I'm stumped. Initially I though I could extend the denominators to "fit them all" but it's a very tedious process. Surely there is a more effective approach! Could you point me in the right direction?
$$\begin{equation}\begin{aligned} \frac{1}{x-1} - \frac{1}{x-2} &= \frac{1}{x-3} - \frac{1}{x-4} \end{aligned}\end{equation}$$
Hint
Simplify each side by combining the fractions
$$\frac{1}{x-1}-\frac{1}{x-2}=\frac{1}{x-3}-\frac{1}{x-4}$$ $$\frac{(x-2)-(x-1)}{(x-1)(x-2)}=\frac{(x-4)-(x-3)}{(x-3)(x-4)}$$ $$\frac{-1}{(x-1)(x-2)}=\frac{-1}{(x-3)(x-4)}$$
Can you take it from here?