Why do you have to check all the 'sections' of an inequality when solving for x? For instance, the inequality
$x^2+3x+2>0$ factors into $(x+2)(x+1)>0$
If this were just an equation (i.e ...=0) you would know the solutions are x={-2, -1}.
But, because it's an inequality, you have to check the intervals. For instance, by drawing a number line and evaluating the inequality expression at each of the intervals and marking whether it satisfies the condition or not. How come the usual method doesn't work?
If you use the inequality symbol is an equals sign and do it the usual way you would get $x>-1$ and $x>-2$ which is not right.

You can alternatively tackle the problem as follows:
\begin{align*}
(x + 2)(x + 1) > 0 & \Longleftrightarrow ((x + 1) + 1)(x + 1) > 0\\\\
& \Longleftrightarrow (x + 1)^{2} + (x + 1) > 0 \\\\
& \Longleftrightarrow (x + 1)^{2} + (x + 1) + \frac{1}{4} > \frac{1}{4}\\\\
& \Longleftrightarrow \left[(x + 1) + \frac{1}{2}\right]^{2} > \frac{1}{4}\\\\
& \Longleftrightarrow \left(x + \frac{3}{2}\right)^{2} > \frac{1}{4}\\\\
& \Longleftrightarrow \left|x + \frac{3}{2}\right| > \frac{1}{2}\\\\
& \Longleftrightarrow \left(x < -2\right)\vee(x > -1)
\end{align*}
Hopefully this helps!
