Square in an Inequality

Say we have the inequality $x^2 \leq 4$. This should imply, unless I'm mistaken, $-2 \leq x \leq 2$.

This sounds rather trivial, but I'm having some trouble understanding why exactly it's the case or what the general rule may be in this case. (A proof would actually be extremely helpful, if anyone has ideas on how to put one together.)

Thanks.