Solve
\[\frac{|x - 13|}{x^2 - 9x - 52} \le 0.\]
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 $x \in \boxed{(-4,13)}.$