# Inequality with absolute values |x+y|/(1+|x+y|) <= |x|/(1+|x|) +|y|/(1+|y|) [duplicate]

$$\frac{|x+y|}{1+|x+y|}\leq \frac{|x|}{1+|x|} +\frac{|y|}{1+|y|}$$

How can i solve this inequality? I have solved it in a long way but i guess there should be an easier way

## marked as duplicate by Arthur, Martin R, Community♦Feb 4 at 14:52

The easy way is to get rid of the absolute values by considering the cases $$x\geq 0$$ and $$x<0$$ for $$|x|$$, $$y\geq 0$$ and $$y<0$$ for $$|y|$$, and $$x+y\geq 0$$ and $$x+y<0$$ for $$|x+y|$$.
Hence, you get $$8$$ cases.