TITLE: Triangle inequality for subtraction?
QUESTION [34 upvotes]: Why is $|a - b| \geq|a| - |b|$?

REPLY [10 votes]: The length of any side of a triangle is greater than the absolute difference of the lengths of the other two sides: 
$$||a|-|b||\leq |a-b|$$
Here is a proof:
$$|a+(b-a)|\leq |a|+|b-a|$$
and,
(1) $$|a-b|\geq |a|-|b|$$
Interchanging $a$ and $b$, we get also
(2)  $$|a-b|\geq |b|-|a|$$
Combining (1) and (2) we get our desired result.