Solving inequalities such as x² − x − 6 > 0 by factorising, finding the critical values, and using a sign table or parabola sketch — with the answer in interval notation.
A quadratic inequality asks where a parabola is above or below the x-axis. The method is always the same four steps.
1The method
- Move everything to one side so the other side is 0.
- Factorise and find the critical values (where it equals 0).
- Decide which regions satisfy the inequality — sketch the parabola or use a sign table.
- Write the answer in interval notation.
Sketch a rough parabola through the two critical values. If a > 0 it opens up, so it is below the axis between the roots and above the axis outside them. That picture answers the question instantly.
2Greater than zero: outside the roots
- 1Factorise: (x − 3)(x + 2) > 0.Critical values are x = 3 and x = −2.
- 2a = 1 > 0, so the parabola opens upwards.It is positive OUTSIDE the roots.
- 3x < −2 or x > 3.Two separate regions — joined by 'or', never 'and'.
- 4Interval notation: (−∞ ; −2) ∪ (3 ; ∞).The union symbol joins the two pieces.
3Less than zero: between the roots
- 1Same factors: (x − 3)(x + 2) ≤ 0.Same critical values.
- 2The parabola is below the axis BETWEEN the roots.
- 3−2 ≤ x ≤ 3, i.e. [−2 ; 3].Square brackets because ≤ includes the endpoints.
Writing the 'greater than' answer as −2 > x > 3 is meaningless — no number is both. When the solution is two separate regions you must use or (∪).
4When the coefficient is negative
- 1Multiply by −1 and FLIP the sign: x² − 4 < 0.Multiplying an inequality by a negative reverses it.
- 2Factorise: (x − 2)(x + 2) < 0.Critical values ±2.
- 3Below the axis, so between the roots: −2 < x < 2.Interval: (−2 ; 2).
Practice exercises
Work each one out, then click to reveal the answer.
- 1Solve x² − x − 6 > 0
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(x−3)(x+2) > 0 → x < −2 or x > 3 - 2Solve x² − x − 6 ≤ 0
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−2 ≤ x ≤ 3, i.e. [−2 ; 3] - 3Solve x² − 9 < 0
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−3 < x < 3 - 4Solve x² − 4 ≥ 0
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x ≤ −2 or x ≥ 2 - 5Solve x² + 5x + 6 < 0
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(x+2)(x+3) < 0 → −3 < x < −2 - 6Solve x² − 5x > 0
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x(x−5) > 0 → x < 0 or x > 5 - 7Solve −x² + 4 > 0
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−2 < x < 2 - 8Write x < −1 or x > 4 in interval notation.
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(−∞ ; −1) ∪ (4 ; ∞)
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