A basic inequality: $a-b\leq |a|+|b|$ 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.

$$
 a - b = a + (-b) \leq |a| + (-b) \leq |a| + |-b| = |a| + |b|
$$
Above, we used the inequality $x \leq |x|$ twice: first with $x=a$ , and then with $x=-b$.
