Do these two equations mean the same thing?

Question

Do the equations

$|x|<a$ and $x<|a|$ have the same solutions , i.e

$-a<x<a$

In general , do the two equations mean the same thing , that the absolute value of $x$ is less that the distance of $a$ from the origin , on either side of the origin ?

Answer 1

If $a=-1$, then$$x<|a|\iff-1<x<1,$$whereas there is no $x$ such that $|x|<a$.

Answer 2

Because $| t|=\max\{t,-t\}$, the inequality $|x|<a$ is equivalent to $$x<a\quad\land\quad -x<a $$ whereas $x<|a|$ is equivalent to $$ x<a\quad\lor\quad x<-a.$$ I suppose tha tthe lack of equivalence becomes obvious from this.