Why is this solution to equation $|f(x)| = g(x)$ also possible?

Question

Given equation $|f(x)| = g(x)$ I understand the first method, but I don't understand the second method (marked grey below) - why is it also equivalent?

Answer 1

Surely there is a typo in your book, it must be if $$f(x)<0$$ then $$-f(x)=g(x)$$

Answer 2

The correct solution is this:

Because $f(x) = g(x)$, $f(x)$ being positive also means $g(x)$ being positive (because of equality!), so more exactly f(x) being non-negative also means g(x) being non-negative(because of equality!).

Likewise, because $-f(x) = g(x)$, $f(x)$ being negative also means $g(x)$ being positive (because of the equality taking into consideration the inversed sign!).

So the second is just another way of writing the first method - in case that it is easier to solve inequalities with $f(x)$, thouogh the first method is more clear and straight-forward!