Why does $\frac{|x|}{x^2}$ reduce to $\frac{1}{|x|}$?

If you want, you can look at is like this: $$\frac{|x|}{x^2} = \frac{|x|}{|x|^2} = \frac{1}{|x|}$$

$$\frac{|x|}{x^2} = \frac{\sqrt{x^2}}{x^2}= \frac{1}{\sqrt{x^2}} = \frac{1}{|x|}$$

$\left|x\right|=x$ if $x>0$ and $\left|x\right|=-x$ if $x<0$

if $x>0$ we will have $\frac{x}{x^2}=\frac{1}{x}$

if $x<0$ we will have $\frac{-x}{x^2}=-\frac{1}{x}$

So, $\frac{\left|x\right|}{x^2}=\frac{1}{\left|x\right|}$.