Prove using the triangular inequality that: $|a+b| \geq |a| - |b|$

Question

How can I prove using the triangular inequality that: $$|a+b| \geq |a| - |b|$$

I already proved it by considering all 8 possible scenarios (like a>b and b=0 ... etc) However I couldn’t manage to find the way to prove it only by using the triangular inequality.

This is the "reverse triangle inequality". See here or here for example. — Minus One-Twelfth, Mar 23 '19 at 00:21

score 6 · Accepted Answer · answered Mar 23 '19 at 00:19

6

Hint: the inequality you wish to prove is equivalent to $$|a + b| + |-b| \ge |a|.$$

answered Mar 23 '19 at 00:19

Theo Bendit

50,900

score 3 · Answer 2 · answered Mar 23 '19 at 00:21

3

By triangular inequality we have: $$|a|=|a+b +(-b)|\leq |a+b|+|b|.$$ Thus, $$|a+b| \geq |a| - |b|.$$

answered Mar 23 '19 at 00:21

S. Maths

2,238

Prove using the triangular inequality that: $|a+b| \geq |a| - |b|$

2 Answers2