Sign of Rational Function

Factoring numerator and denominator: $f(x) = \frac{(x+5)(x-3)}{(x-9)(x+2)}$

Critical points in increasing order are $-5, -2, 3, 9$. Checking signs on intervals:

• For $x > 9$: Positive
• For $3 < x < 9$: Negative
• For $-2 < x < 3$: Positive
• For $-5 < x < -2$: Negative
• For $x < -5$: Positive

Thus, $f(x) < 0$ when $-5 < x < -2$ or $3 < x < 9$.