Inequality with absolute values |x+y|/(1+|x+y|) <= |x|/(1+|x|) +|y|/(1+|y|)

Question

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

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

Answer 1

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.