Trait proof with absolute value

Question

Will be happy if you guys check my proof. I have to prove that: $$ |x| < m \iff -m < x < m$$ so I'm going back to the basic definition of the absolute value: $$ x < 0 \implies |x| = -x$$ $$ x > 0 \implies |x| = x$$ considering $x$ in both cases and createing to inequalities: $$-x < m \implies x > -m $$ $$ x < m$$ now from the two inequalities above, concluding about the range of $x$ : $$-m<x<m$$ thanks in advance :)

Answer 1

Your proof is essentially correct. I would suggest just a little adjustment.

1) For $x<0$ you got $x>-m$ so you have $-m<x<0$;

2) For $x\ge 0$ you got $x<m$ so you have $0\le x<m$;

Putting both together you get:

$$-m<x<m$$