Is it safe to use $|f(x)|Some peoples often use the following.
$$|f(x)|<g(x)\iff -g(x)<f(x)<g(x)$$
The weird part occurs when $g(x)<0$, for example:
$$10<f(x)<-10\to |f(x)|<-10$$
where $x\in\{\}$  for both sides of implication which is  true.
Is it safe to use $|f(x)|<g(x)\iff -g(x)<f(x)<g(x)$?

It is correct. If you have $|f(x)|<g(x)$, then $g$ must be positive everywhere since $0 \le |f(x)|$. Conversely, if $-g(x)<f(x)<g(x)$, the again $g$ must be positive everywhere because we have $-g(x)<g(x)$ which is impossible if $g(x) \le 0$.
