Quadratic Inequalities
Quadratic Inequalities and Number Lines
A quadratic inequality asks where a quadratic expression is positive, negative, or equal to zero. Its roots divide the number line into intervals that you can test.
Intervals and endpoints
Use open endpoints for < or > because the roots are not included. Use closed endpoints for ≤ or ≥ when equality is allowed.
How to solve a quadratic inequality
- Move all terms to one side.
- Factor and find the roots.
- Mark the roots on a number line.
- Determine the sign in each interval.
- Choose the intervals that satisfy the inequality.
Worked example
Solve x² − 1 < 0
- Factor: (x − 1)(x + 1) < 0
- The roots are −1 and 1.
- The upward-opening quadratic is negative between its roots.
- The inequality is strict, so the endpoints stay open.
The solution is −1 < x < 1, or (−1, 1).
Common mistakes
- Including roots in a strict inequality.
- Choosing the interval without checking the sign.
- Finding the roots but not writing an interval.
Practice more
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