I am supposed to find the range of the following. At present I do not know how to get started. Obviously as is the case with numerous symmetrical expressions, one of the extrema occur when all variables are equal, so $3$ would be a value of extrema, but how to prove such an argument rigorously is non-obvious. Any hints would be appreciated. Thanks.
$$\sum_{cyc}\frac{|x+y|}{|x|+|y|} \text{ for }x,y,z\in\mathbb{R}- \{0\}$$
(Fill in the gaps as needed. If you're stuck, show your work and explain why you're stuck.)
Hint: If $xy \geq 0$, show that $ \frac{ | x+y | } { |x| + |y| } = 1$.
If $ xy < 0$, show that $ 0 \leq \frac{ | x+y | } { |x| + |y| } < 1 $.
Hence, conclude that
$$ 1 \leq \sum \frac{ | x+y | } { |x| + |y| } \leq 3. $$