TITLE: Show that $|a+b|=|a|+|b|\Leftrightarrow ab\geq0$
QUESTION [3 upvotes]: Assume $a, b \in \mathbb{R}$. Show that
$$|a+b|=|a|+|b|\Leftrightarrow ab\geq0$$
The triangle inequality says that for $a,b\in\mathbb{R}$, $|a+b|\leq |a|+|b|$.
So I belive I can say 
$$|a+b|-|a|-|b|\leq 0$$
I am not quite sure where to go from there, though, or if I am on the right track.

REPLY [9 votes]: By squaring the two sides,
$$ |a+b|=|a|+|b|\Leftrightarrow a^2 + 2ab + b^2 = a^2 + 2|ab| + b^2.$$
Thus 
$$ |a+b|=|a|+|b|\Leftrightarrow ab = |ab| \Leftrightarrow ab\geq 0.$$
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