Quadratic Inequalities
Solving x² − 5x + 6 < 0 looks like solving an equation, and the first half is. The second half, deciding which side of the roots you want, is where marks are won or lost. A quick sketch of the parabola makes it impossible to get wrong.
1The method: solve, sketch, read off
- Rearrange so one side is 0: e.g. x² − 5x + 6 < 0.
- Find the critical values by solving the equation x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3.
- Sketch y = x² − 5x + 6: a ∪-shaped parabola crossing the x-axis at 2 and 3.
- “< 0” means below the x-axis. That is the part between the roots.
Key ideaFor a ∪-shaped graph (positive x² term):
• < 0 or ≤ 0 → between the roots: one inequality, a < x < b.
• > 0 or ≥ 0 → outside the roots: two inequalities, x < a or x > b.
• < 0 or ≤ 0 → between the roots: one inequality, a < x < b.
• > 0 or ≥ 0 → outside the roots: two inequalities, x < a or x > b.
2“Greater than” gives two regions
Worked example
Solve x² − x − 12 ≥ 0.
- Factorise: (x − 4)(x + 3) ≥ 0. Critical values: x = −3 and x = 4.
- We want the graph on or above the x-axis, which is outside the roots.
- Because of “≥”, the critical values are included.
Don’t do this
- Writing −3 ≥ x ≥ 4. This says x is less than −3 and greater than 4 at the same time, which is impossible. Write the two parts separately with “or”.
3Rearrange before you factorise
Worked example
Solve x² < 2x + 15.
- Get zero on one side: x² − 2x − 15 < 0.
- Factorise: (x − 5)(x + 3) < 0. Critical values −3 and 5.
- “< 0”, so between the roots.
Worked example
Solve 6 + x − x² > 0.
- The x² term is negative, so first multiply by −1 and reverse the inequality: x² − x − 6 < 0.
- Factorise: (x − 3)(x + 2) < 0, so the critical values are −2 and 3.
- “< 0” → between the roots.
Exam tipMultiplying or dividing an inequality by a negative number flips the sign. The other way is to sketch the ∩-shaped graph directly. Either method gives the same answer.
4When it won’t factorise
Worked example
Solve x² + 4x − 1 < 0, giving your answer in surd form.
- Solve x² + 4x − 1 = 0 by completing the square: (x + 2)² − 5 = 0, so x = −2 ± √5.
- “< 0”, so between the roots.
5Number lines, set notation and integer solutions
You may be asked to show the answer on a number line, write it in set notation, or list the integers that satisfy it.
In set notation: {x : 2 < x < 3}, or for two regions {x : x ≤ −3} ∪ {x : x ≥ 4}.
Worked example
Find the integers that satisfy both x² < 20 and 3x − 2 > 1.
- x² < 20 → −√20 < x < √20, and √20 ≈ 4.47. Integers: −4, −3, …, 3, 4.
- 3x − 2 > 1 → 3x > 3 → x > 1.
- Integers in both lists: 2, 3, 4.
6Quadratic inequalities in context
Worked example
A rectangle has width x cm and length (x + 4) cm. Its area must be less than 60 cm². Find the possible values of x.
- Area: x(x + 4) < 60 → x² + 4x − 60 < 0.
- Factorise: (x + 10)(x − 6) < 0, so −10 < x < 6.
- A length must be positive, so x > 0.
Key ideaIn context, always ask whether negative values make sense, and cut the answer down if they don’t.
Common mistakes
- Stopping after finding the critical values. x = 2 or x = 3 is not the answer to an inequality.
- Choosing “between” when it should be “outside” (or vice versa). Sketch the graph every time.
- Forgetting to flip the sign when multiplying or dividing by a negative.
- Writing two-region answers as one combined inequality, like −3 ≥ x ≥ 4.
- Mixing up strict (<, >) and inclusive (≤, ≥) signs, or using open/closed circles wrongly.
★Key Vocabulary Recap
- Critical values
- The solutions of the related equation; the boundaries of the solution set.
- Strict inequality
- < or >: the boundary value is not included.
- Inclusive inequality
- ≤ or ≥: the boundary value is included.
- Set notation
- Writing a solution as a set, e.g. {x : −2 < x < 3}.
Specification reference (Pearson Edexcel 4XMA1, Unit 2 Higher)
- Solve quadratic inequalities in one unknown and represent the solution set on a number line.
- Identify harder examples of regions defined by linear inequalities (see the Inequalities notes).
Next stepReady to practise? Try the 12 exam-style questions, then check your working against the worked solutions.
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