How to solve this inequality question without manual checking? Question:

Find the maximum integral value which satisfies: $$\frac{x-2}{x^2-9}<0$$

I know that this means either of the following:
#1. $x-2<0$ and $x^2-9>0$. Implies that $x \in (3, 2)$, which is false. [1]
#2. $x-2>0$ and $x^2-9<0$. Implies that $x \in (2, 3)$, which is true. [1]
However, this does not give the correct answer. Of course, I can solve this by checking values, but I want a proper method of solving.
I, thus, need help in figuring out the method to get the correct answer
[1] - $x^2-9>0 \Rightarrow x^2>9 \Rightarrow x>3$

Because of the denominator, $|x|=3$ is forbidden. 
If $|x|<3$, we need $x-2>0$, which is impossible for integer $x$.
If $|x|>3$, we need $x-2<0$, which together with $|x|>3$ implies that $x\le -4$ (integer!), so  $x=-4$ is the answer.
