Absolute value of x²-1 greater than or equal to 3 (|x²-1| ≥ 3)

Statement

Solve the inequality $|x^2-1|\ge3$.

Solution

It is of the type $|A|\ge B$ with $B=3>0$: it is equivalent to $A\ge B\ \lor\ A\le-B$, that is $x^2-1\ge3\ \lor\ x^2-1\le-3.$

• $x^2-1\ge3 \iff x^2\ge4 \iff x\le-2\ \lor\ x\ge2$;
• $x^2-1\le-3 \iff x^2\le-2$: impossible.

Solution: $\boxed{x\le-2\ \lor\ x\ge2}$.

Links

Topics: Parabola
Concepts: Absolute value inequalities · Absolute value
Skills: Solving inequalities
Exercise type: Solving an inequality