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.