TITLE: A basic inequality: $a-b\leq |a|+|b|$
QUESTION [8 upvotes]: Do we have the following inequality:
$$a-b\leq |a|+|b|$$
I have considered $4$ cases: 

$a\leq0,b\leq0$
$a\leq0,b>0$
$a>0,b\leq0$
$a>0,b>0$

and see this inequality is true. However I want to make sure about that.

REPLY [4 votes]: Right. Observe that $a\leq |a|$ and that $(-b)\leq |(-b)|=|b|.$ Adding, we have $a-b=a+(-b)\leq |a|+|(-b)|=|a|+|b|.$