I recently encountered this algebra transformation from an absolute-value sign to plus-minus sign:
$$|y|=5e^{2x-2}\iff y=\pm5e^{2x-2}$$
I am unable to get my head around how it really works. What is the underlying principle that justifies such transformation?
On
Adding to the accepted answer:
$$|y|=5e^{2x-2}\iff y=\pm5e^{2x-2}$$
In general, $$|y|=f\quad\iff\quad y=\pm f\:\:\text{and}\:\:\color{brown}{f\ge0}.$$
The condition $\color{brown}{f\ge0}$ is important: for example, $$y=\pm2x\kern.6em\not\kern-.6em\implies|y|=2x.$$
In your given example, $f=5e^{2x-2}\ge0.$
If $|x|=1$, then it is either $x=1$ or $x=-1$.
Hence if $|x|=y$, then it is possible that $x=y$ or $x=-y$.