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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.

1 Answers 1