I am unsure if have solved the following inequality correctly:
$ \dfrac{2x+3}{x+5} \leq \dfrac{x+1}{|x-1|} \tag{1}$
I've proceeded as follows.
If $x>1$ then $|x-1|=(x-1)$
If $x<1$ then $|x-1|=-x-1$
I've then solved for those seperate inequalities,
$\dfrac{2x+3}{x+5} \leq \dfrac{x+1}{x-1}$
$\dfrac{2x+3}{x+5} \leq \dfrac{x+1}{-x-1}$
The problem is that the union of their solution intervals yields a different result from the inequality (1) when I enter it into Wolfram Alpha. I am afraid I have not solved it correctly.