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Let $a$ and $b$ in $\mathbb{R}$

1) Show that $||a|-|b||\leq|a+b|\leq|a|+|b|$.

2) Prove that the one or the other of the two inequalities is an equality.

It's fine whit the 1st question but i can't figure out the 2nd.

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1 Answer

up vote 6 down vote accepted

The equality $ |a+b|= |a| + |b|$ holds if $a$ and $b$ have the same sign. If they have opposite signs, then without loss of generality suppose $a\geq 0 $ and $b<0.$ Then since $|a|=a$ and $|b| = -b,$ $|a|-|b| = a+b $ so the left inequality is an equality.

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