### Instruction: Solve
\[\frac{|x - 13|}{x^2 - 9x - 52} \le 0.\].### Response: To solve an inequality involving absolute value, I need to consider two cases: one where the expression inside the absolute value is positive, and one where it is negative.
I also need to be careful about the denominator, because it cannot be zero.
So I should factor it and find its zeros first.
The denominator can be factored as \[(x^2 - 9x - 52) = (x - 13)(x + 4).\]
So the denominator is zero when \[x = 13 \text{ or } x = -4.\]
These values divide the number line into three intervals: \[(-\infty, -4), (-4, 13), (13, \infty).\]
I will test each interval separately for the two cases of the absolute value.
Case 1: If \[x - 13 \ge 0,\] then \[|x - 13| = x - 13.\]
The inequality becomes
\[\frac{x - 13}{(x - 13)(x + 4)} \le 0.\]
I can cancel out the common factor of \(x - 13\) from the numerator and denominator, as long as I remember that \(x \ne 13\).
The inequality simplifies to
\[\frac{1}{x + 4} \le 0.\]
This is true when \[x + 4 < 0\] or \[x < -4.\]
However, this contradicts the assumption that \[x - 13 \ge 0.\]
So there is no solution in this case.
Case 2: If \[x - 13 < 0,\] then \[|x - 13| = -(x - 13) = 13 - x.\]
The inequality becomes
\[\frac{13 - x}{(x - 13)(x + 4)} \le 0.\]
This is a rational function with a negative leading coefficient, so it has opposite signs at the end intervals and the same sign at the middle interval.
To find out which sign is which, I can plug in any value in each interval and check the sign of the function.
For example, if I plug in \(x = -5\) in the first interval, I get
\[\frac{13 - (-5)}{(-5 - 13)(-5 + 4)} = \frac{18}{(-18)(-1)} = 1 > 0.\]
So the function is positive in the first interval \((-\infty, -4)\), and by symmetry, it is also positive in the last interval \((13, \infty)\).
Therefore, it is negative in the middle interval \((-4, 13)\).
Since I want the function to be less than or equal to zero, the solution is the middle interval, excluding the endpoints where the denominator is zero.
So the final answer is \[-4 < x < 13.\]