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

Question

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.

Answer 1

Use the triangle inequality: $$ a - b \leq \vert a - b \vert \leq \vert a \vert + \vert b \vert. $$

Answer 2

$$ a - b = a + (-b) \leq |a| + (-b) \leq |a| + |-b| = |a| + |b| $$

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

Answer 3

Right. Observe that $a\leq |a|$ and that $(-b)\leq |(-b)|=|b|.$ Adding, we have $a-b=a+(-b)\leq |a|+|(-b)|=|a|+|b|.$