jmc

We can factor the denominator to get $$\frac{|x-13|}{(x+4)(x-13)}\le 0.$$Note that $|x-13|\ge 0$ for all real numbers $x.$

If $x<-4,$ then $x+4<0$ and $x-13<0,$ so the inequality is not satisfied.

If $-4<x<13,$ then $x+4>0$ and $x-13<0,$ so the inequality is satisfied.

If $x>13,$ then $x+4>0$ and $x-13>0,$ so the inequality is not satisfied.

The solution is then $x\in \boxed{(-4,13)}.$