For every $a, b, x \in \mathbb{R}$ such that $a \le x \le b$, we have $|x| \le \max\{|a| , |b|\}$

Question

I was given an assigmnent in my Infintesiaml math course, and I'm having trouble proving the statement below:

For every $a, b, x \in \mathbb{R}$ such that $a \le x \le b$, we have $|x| \le \max\{|a| , |b|\}$

Hint: denote $r = \max \{|a| , |b|\}$.

Could it be false? I have tried using the triangle inequality, with and without the mentioned hint, and also tried isolating $x$ and $a$, or $x$ and $b$, but i think there is something important that I am missing.

I saw a similar question in a different post but the answers there are not explained very well.

Answer 1

Hint:

Remember that for any $u$, $|u|=\max(u,-u)$. Therefore if $r=\max\bigl(|a|,|b|\bigr)$, we have $\;-r \le a,\,b,-a,-b\le r$. What can you deduce for $\max(x,-x)$?