TITLE: Triangle inequality applied to subtraction?
QUESTION [10 upvotes]: So I checked out other questions and it appears my specific question wasn't answered
I was wondering if the following is valid:
$$
\lvert a - b \rvert \le \lvert a \rvert + \lvert -b \rvert = \lvert a \rvert + \lvert b \rvert \\
\therefore \; \lvert a - b \rvert \le \lvert a \rvert + \lvert b \rvert
$$
Thanks in advanced!

REPLY [10 votes]: Yes, that's valid. You may also be interested in the reverse triangle inequality.
Unfortunately, the reverse triangle inequality does nothing to reverse triangle inequality:

(from Brown Sharpie)