Is this a typo in my math book - incorrect modulus operation?

Question

In the second line (I marked it with an arrow) I think it should be $g(x)<0$

Answer 1

If we have $$ |f(x)| + |g(x)| = f(x) - g(x) $$ then $g(x) = 0$ is possible. It would be wrong to write $g(x) < 0$ and still have "$\Leftrightarrow$" between the two statements as they would no longer be equivalent.

Yes, by definition you may have $$ |g(x)| = \cases{g(x) & if $g(x)\geq 0$\\-g(x) & if $g(x)<0$} $$ But this still means that $$ |g(x)| = -g(x)\iff g(x)\leq 0 $$ as $0 = -0$

Answer 2

You could use $<$ rather than $\leq$ and the definition would be equivalent, as, in the case of $g(x)=0$, then $g(x)=-g(x)=0$, so it doesn't matter either way. You would still have $$|f(x)| + |g(x)| = |f(x)|$$, or

$$|f(x)| + |g(x)| = |f(x)| + |g(x)|$$ , or $$|f(x)| + |g(x)| = |f(x)| - |g(x)|$$

They are all the same thing!