I am working on some absolute equation problems like the following:
$$\begin{align}
& {|x-4|} \lt 1 \\
& 1 \le |x| \le 4 \\
& |x+3| = |2x+1|
\end{align}$$
Now, for both of these equations, I simply squared both sides to get rid of the absolute and then continued solving from there. Now my question is: when can I not do this and what is the alternative if I can't?
Thanks a bunch!
Best Answer
Order of a relation is preserved, when you apply a strictly monotonously increasing function (for $\leq, \geq$ you can drop the strictly). $$f: x \mapsto x^2$$ is strictly monotonously increasing on $[0, \infty)$. so you can square whenever all expressions are guaranteed to be $\geq 0$.