Why if $a<b$ and $-a<b$ we can say that $|a|<b$?

Question

Why if $a<b$ and $-a<b$, then we can say that $|a|<b$?

Maybe this is trivial by I don't know how to proof it.

Answer 1

If $a \geq 0$, then $|a| = a$; so $a < b$ implies $|a| < b$.

If $a < 0$, then $|a| = -a$; so $-a < b$ implies $|a| < b$.

Answer 2

We have either $|a|=a$ or $|a|=-a$, hence what you wrote holds

Answer 3

Suppose not, and show it contradicts your hypotheses.

Note that $|a|$ is either $a$ or $-a$.

In the first case, you would have $a\geq b$, contrary to the hypothesis $a<b$.

In the second case, you would have $-a\geq b$, contrary to the hypothesis that $-a<b$.

In either case, the supposition that the result is false leads to a contradiction of the hypotheses. Therefore, the hypotheses imply that the result is true, quod erat demonstrandum.

Answer 4

One option:

Note that $|a| = \max(a, -a)$. Hence if $a < b$ and $-a < b$, then

$$\max(a, -a) < b$$

That is

$$|a| < b$$