I have to solve different types of inequalites of this type:
$$|y^2-y-2|\geq 4 + |y^2+y-2|+|y+4|+|y|$$
I know the standard method for solving these inequalities, by finding the all zeros of the expressions and then analysing the sign of the expressions inside the absolute values for all the regions resulting from the zeros. But is there a smarter way to solve similar inequalities?
EDIT: The method can also use calculators but no advanced calculators (with plotting abilities or something like that.
When there is special structure, you can be clever. In your example, just saying $|y^2-y-2|\gt 4+ |y^2+y-2|$ forces $y$ to be greater than $2$ in absolute value. That means $y$ must also be negative to make the left side larger. That allows you to remove the absolute value bars on $y$ at the end, and to know that the long expressions are positive, leaving $y^2-y-2\geq 4 + y^2+y-2+|y+4|-y$ and $y \lt -2$. Now you can transform this to $-y \ge 4+|y+4|$ and $y+4$ must be negative, so we have $-y \ge 4$ or $y \le -4$.