### Instruction: Solve
\[\frac{|x - 13|}{x^2 - 9x - 52} \le 0.\].### Response: To solve an inequality involving absolute value, I can split it into two cases: one where the expression inside the absolute value is positive, and one where it is negative.
So, I first need to find the value of x that makes x - 13 equal to zero, which is x = 13.
Then, I can consider the two cases: x - 13 > 0 and x - 13 < 0.
For the first case, x - 13 > 0, I can drop the absolute value bars and write

\[\frac{x - 13}{x^2 - 9x - 52} \le 0.\]
To simplify this inequality, I can factor the denominator as (x - 13)(x + 4), since -13 and 4 are factors of -52 that add up to -9.
Then, I get

\[\frac{1}{x + 4} \le 0.\]
This inequality is true when x + 4 is negative, that is, when x < -4.
However, I also need to remember that x - 13 > 0, so I have to take the intersection of these two conditions.
That means x has to satisfy -4 > x > 13, which is impossible.
Therefore, there are no solutions in this case.
For the second case, x - 13 < 0, I can write

\[\frac{-(x - 13)}{x^2 - 9x - 52} \le 0.\]
Again, I can factor the denominator as (x - 13)(x + 4) and simplify the inequality as

\[\frac{-1}{x + 4} \le 0.\]
This inequality is true when x + 4 is positive, that is, when x > -4.
I also need to keep in mind that x - 13 < 0, so x < 13.
Therefore, x has to satisfy -4 < x < 13.
This is the solution for this case.
Since there are no solutions in the first case, the final answer is the solution in the second case.