Q:

What is the right approach to use for factoring a rational inequality

Here is an example:
$\frac{x^2 - 3x + 2}{x + 1} -5 > 0$
My approach would be to factor, find the undefined areas and the zeros, and then pick some points in the intervals left to see what I find. I'm not really sure if that's the right way to go, and to find the zeros I would just throw an equals sign in instead of the inequality and find the roots. Is this the right approach, and are there any techniques to improve it?

A:

This is not a general approach, but in this example $x=-1$ is 'special', so split the consideration into $I_- = (-\infty,-1)$ and $I_+ = (-1, +\infty)$.
On $I_+$, we have $x^2 - 3x + 2 - 5 (x + 1) = x^2-8x-3 > 0$. The factors are $4 \pm \sqrt{19}$, hence if $x \in I_+$, then $\frac{x^2 - 3x + 2}{x + 1} -5 > 0$ iff $x > 4 + \sqrt{19}$ or $x < 4 - \sqrt{19}$.
On $I_-$, we have $x^2 - 3x + 2 - 5 (x + 1) =x^2-8x-3 < 0$. From the previous reasoning, we have that if $x \in I_-$, then $\frac{x^2 - 3x + 2}{x + 1} -5 \leq 0$.
Hence $\frac{x^2 - 3x + 2}{x + 1} -5 > 0$ iff $ x \in (-1,4 - \sqrt{19}) \cup (4 + \sqrt{19}, +\infty)$.

