General splitting method

The basic method for an inequality with an absolute value is the same as for equations: you split it into the two cases according to the sign of the argument. It always works, even when the shortcuts for the special cases cannot be applied.

In brief — General method

Given an inequality of the type $\bigl[\text{espressione con }|A(x)|\bigr]\gtrless0,$ you split it into the two cases:

1. Case (a): $A(x)\ge0$. In this region the absolute value is removed without changing sign, that is $|A(x)|=A(x)$: you solve the expression with $A(x)$ in place of $|A(x)|$ and intersect the solution with the condition $A(x)\ge0$.
2. Case (b): $A(x)<0$. Here $|A(x)|=-A(x)$: you solve with $-A(x)$ in place of $|A(x)|$ and intersect with $A(x)<0$.

The final solution is the union of the solutions of the two cases: $S=S_a\cup S_b$.

Diagram of the splitting into the two cases and the union of the solutions.

Links

Topics: Parabola
Concepts: Inequalities with absolute value · Absolute value
Skills: Reasoning by cases · Solving inequalities