Proving an equivalence relation in regards to sum of absolute values.

Question

Prove that the relation R is an equivalence relation on the set of real numbers.

$$(x,y) \in R \iff |x+y| = |x| + |y|$$

I did prove the reflexivity as well as the symmetry, but I am stuck on how to prove the transitivity of this relation.

Answer 1

It's not an equivalence relation. Hint: What things are related to $0$.

Answer 2

Notice that $1\sim0$ and $0\sim-1$, but $1\not\sim -1.$