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.

  • $\begingroup$ What's a more wordy way of saying $ab \geq 0$ ? $\endgroup$ – pjs36 Sep 18 '15 at 2:27

By squaring the two sides,

$$ |a+b|=|a|+|b|\Leftrightarrow a^2 + 2ab + b^2 = a^2 + 2|ab| + b^2.$$

Thus $$ |a+b|=|a|+|b|\Leftrightarrow ab = |ab| \Leftrightarrow ab\geq 0.$$

  • $\begingroup$ I love when these things don't become case-seeking. +1 $\endgroup$ – Aloizio Macedo Sep 18 '15 at 3:36

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