TITLE: What happens when square root is performed in inequalities?
QUESTION [7 upvotes]: Simplify: $x^2 > 1$.
My solution: Taking square root on both sides:
$±x > ±1$
So my results are:

$x > 1$
$x > -1$
$-x > 1$ $\implies$ $(-1 > x)$
$-x > -1$ $\implies$ $(1 > x) $

But I strongly feel this is wrong. What is wrong here?
A step-by-step explanation will help me.

REPLY [5 votes]: You can just go from $x^2 > 1$ directly to $|x| > |1|$. 
Now obviously $|1| = 1$, so $|x| > 1$, therefore either $x > 1$ or $x < -1$.