I have been taught that if $|f(x)| < a$, where $a$ is positive, then simply $-a< f(x) <a,$ and this makes much sense to me. My query is what if I have something like $$|f(x)| < g(x)$$ Do I need to check when is $g(x)<0$ and exclude all those solutions? Or can I ignore checking it and simply go for $-g(x)< f(x) < g(x)$ Here I am taking $g(x)$ a non constant function whose range includes both negative and positive values. I tried to find examples by making my own questions where if I don't check for $g(x)<0,$ I get the wrong answer, but I couldn't frame any.
I'll give an example
Suppose |x²-3x+2| < x+1 Why don't I need to check x+1<0 and remove those solns from my answer? I can directly go -x-1 < x²-3x+2 < x+1 without caring for x+1.
Edit: I wasn't able to explain my query well. Kindly refer to Inequality query related to absolute value function I have reframed my question.
You can ignore it and simply go for $-g(x)< f(x) < g(x)$. Because if $g(x)<0$ then in $-g(x)< f(x) < g(x)$ you have a positive number on the left and negative on the right. And so it cannot have a solution anyway. This is even true if $g(x)\leq 0$. Or equivalently: if $-g(x)< f(x) < g(x)$ has a solution that $g(x)>0$ has to hold.