Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|cite|improve this question

1 Answer 1

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.