Why in module inequality $|f(x)| < A$ with a positive A we do not explicitly check that $f(x)>=0$ and $f(x)<0$?

Question

$|f(x)| < A$ when $А > 0$ is equivalent to the following system:

$$\begin{cases} f(x) < A \\ f(x) > -A \\ \end{cases} $$

Sorry to ask this, but I do not completely understand, why we do not exclude those $x$, for which $f(x)<0$ in the first equation, and $x$ for which $f(x)<0$ in the second equation - I mean why we do not write explicitly two systems of inequalities, uniting their solutions like this:

$$\begin{cases} f(x) < A \\ f(x) >= 0 \\ \end{cases} $$

$$\begin{cases} f(x) > -A \\ f(x) < 0 \\ \end{cases} $$

Answer 1

The two systems are completely equivalent.

In the first you find directly all solutions by intersection of two inequalities, in the second case by union of two different systems of inequalities.