That's the absolute value definition.
yeah. we often take this fact for granted around here. it's hard to come up with a rigorous proof. but here is a possible one for the reverse implication.
Recall that
Thus, by the definition of :
or
or
combining these two inequalities, we obtain:
now try to go the other way. i would probably try to reverse this exact proof. so split the inequality into two, and work on each case by case (if x > 0 or x < 0)