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 $|f(x)|<g(x)\iff -g(x)<f(x)<g(x)$?

For each $f(x)$ and negative $g(x),$ $$|f(x)|<g(x)$$ and $$-g(x)<f(x)<g(x)$$ are both false. Hence, by deductive explosion, $$\forall f \;\;\forall g<0 \bigg(|f(x)|<g(x)\; \iff \; -g(x)<f(x)<g(x)\bigg).$$

Also for each $f(x)$ and negative $g(x),$ $$|f(x)|> g(x)$$ and $$f(x)<-g(x) \;\text{ or }\; f(x)> g(x)$$ are both true. Hence, since a true consequent automatically means a true implication, $$\forall f \;\;\forall g<0 \bigg(|f(x)|> g(x)\;\iff \; f(x)<-g(x) \;\text{ or }\; f(x)> g(x) \bigg).$$
