Why must absolute value be used in this equation? I have a simple equation:
$(x-2)^2 < 3$
My first solution was:
$x-2 < \sqrt{3}$
But this gives me only:
$x < \sqrt{3} + 2$
Which is only one solution, so it's not enough. I've figured it out that I must use:
$|x-2| < \sqrt{3}$
Then, solutions are:
$-\sqrt{3} + 2 < x < \sqrt{3} + 2$
My question is how can I know that here:
$(x-2)^2 < 3$
I must use absolute value after square rooting both sides of inequality? Like: $|x-2| < \sqrt{3}$

If you are not comfortable using $|x-2|$ then you can divide your derivation into two cases as follows:
Case 1: $x-2\ge 0$ :
$(x-2)^2 < 3 \\ \Rightarrow 0 \le x-2 < \sqrt 3 \\ \Rightarrow 2 \le x < 2+\sqrt{3}$
Case 2: $x-2< 0$ :
$(x-2)^2 < 3 \\ \Rightarrow -\sqrt 3 < x-2 < 0\\ \Rightarrow 2-\sqrt 3 < x < 2$
Taking the union of the intervals from the two cases, we get $2-\sqrt{3}<x<2+\sqrt{3}$. The case by case approach is useful for more complex inequalities e.g. $2 < (x-2)^2<3$.
