Let $x,y \in \mathbb{R}$, s.t. $y>0$. Then $|x|<y \iff -y < x < y$.
I'm confused about how to join the cases in this proof into a single interval.
By definition, $|x|= x$ if $x \geq0$ and $|x|=-x$ if $x<0$. So the proof lends itself to two cases.
Case 1: $x \geq 0$
Then $|x|<y \implies x<y$
Case 2: $x < 0$
Then $|x|<y \implies -x<y \implies x>-y$
But the cases are joined by an "or" statement, as opposed to an "and" statement, correct?
So it would be wrong to conclude that the solution lies in the intersection. That is, I cannot immediately conclude $-y<x<y$.
So, what am I missing?
$$|x|<y\implies x\le \max (x,-x)=|x|<y\implies x<y.$$ $$|x|<y\implies -x\le \max (x,-x)=|x|<y\implies -x<y\implies x>-y.$$