Help with showing that $|x+y|=|x|+|y| \longleftrightarrow xy\ge0 $

Question

I need to prove that $|x+y|=|x|+|y| \longleftrightarrow xy\ge0 $.

I proved before that for any $x,y\in R$ holds $|x+y|\le|x+y|$ and I thought maybe it could help me with my arguments to show what I need. On the other hand if $xy\ge0$ then it means that both $x,y$ are positive or negative. I don't see exactly how it helps me somehow.

Any hints on how should I proceed?

Answer 1

Square (note that both sides are non-negative!):

$$|x+y|=|x|+|y|\iff x^2+2xy+y^2=x^2+2|xy|+y^2\iff $$

$$xy=|xy|\stackrel{\text{by def.}}\iff xy\ge 0$$