Need help in solving inequality of this type: $|ax + b| > -c$ This is the equation: $|3x+6|>-12$. I solved it under two cases ($3x+6>-12$) and ($3x+6<12$). More than the answer, what I need to know is - Have I rightly constructed those $2$ cases, if wrong, please explain. Thanks.

Note that $|3x+6|$ is always non-negative. So, $|3x+6|\gt -12$ holds for every $x\in\mathbb R$.

For $|ax+b|\gt -c$ in the title,

* If $c=0$, then it always holds except $x=-\frac ba$ (for $a\
ot=0$).

* If $-c\lt 0$, then it always holds. (your case is included here)

* If $-c\gt 0$, we have $$|ax+b|\gt -c\iff ax+b\lt -(-c)\ \ \ \text{or}\ \ \ ax+b\gt -c$$




* * *

For $C\gt 0$, we have

$$|x|\gt C\iff x\lt -C\ \ \ \text{or}\ \ \ x\gt C$$ $$|x|\lt C\iff -C\lt x\lt C$$