So I want to prove that and its counterpart. I have that
\begin{align*} |x| \leq a \Longleftrightarrow -a \leq x \leq a \end{align*}
$x \leq |x|$, and $|x| \leq a$, so for transivity, $a \geq x$, and then, $-a \leq -|x|$, $-|x| \leq x$, and $-a \leq x$.
The thing is I don't know how to prove from right to left with this one \begin{align*} |x|\geq a \Longleftrightarrow (x\leq -a)\vee(x \geq a) \end{align*}
Can someone tell me a hint, or at least the next step?