Quadratic inequalities

Gold-standard guide

26 mins

What you will learn

Find ranges of values that satisfy a quadratic inequality.

Use a clear step-by-step method for quadratic inequalities.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier algebra skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Sketch the parabola: find the roots, then decide which region satisfies the inequality

Step 1

Factorise the quadratic to find the roots

x² + x − 6 = (x + 3)(x − 2) = 0, so roots are x = −3 and x = 2

Step 2

Sketch the parabola

Since the coefficient of x² is positive (+1), the parabola is U-shaped (opens upward)

Step 3

Identify the region where the graph is ABOVE the x-axis (y > 0)

The U-shape is above the x-axis to the LEFT of −3 and to the RIGHT of 2

Watch out

Students write −3 < x < 2 (the region BETWEEN the roots) when the inequality is > 0 and the parabola opens upward

f

Root method

Find the roots, sketch the parabola, then choose the required region.

f

Upward parabola

$If x^{2} coefficient is positive, the graph is above zero outside the roots.$

Worked example

Solve x² + x − 6 > 0

1

Factorise the quadratic to find the roots: x² + x − 6 = (x + 3)(x − 2) = 0, so roots are x = −3 and x = 2.

2

Sketch the parabola: Since the coefficient of x² is positive (+1), the parabola is U-shaped (opens upward).It crosses the x-axis at −3 and 2.

3

Identify the region where the graph is ABOVE the x-axis (y > 0): The U-shape is above the x-axis to the LEFT of −3 and to the RIGHT of 2.

4

Write the solution: x < −3 or x > 2. Note: two separate regions, connected by 'or'.

Final answer

x < −3 or x > 2

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Solve x² + x − 6 > 0

4 marks4 minsquadratic-inequalities-worked

Show solution

Worked solution

1.Factorise the quadratic to find the roots: x² + x − 6 = (x + 3)(x − 2) = 0, so roots are x = −3 and x = 2.
2.Sketch the parabola: Since the coefficient of x² is positive (+1), the parabola is U-shaped (opens upward).It crosses the x-axis at −3 and 2.
3.Identify the region where the graph is ABOVE the x-axis (y > 0): The U-shape is above the x-axis to the LEFT of −3 and to the RIGHT of 2.
4.Write the solution: x < −3 or x > 2. Note: two separate regions, connected by 'or'.

Final answer

x < −3 or x > 2

Mark points

M1: factorise the quadratic to find the roots
M1: sketch the parabola
M1: identify the region where the graph is above the x-axis (y > 0)
M1: write the solution
A1: x < −3 or x > 2

Watch out

Students write −3 < x < 2 (the region BETWEEN the roots) when the inequality is > 0 and the parabola opens upward.That region is actually below the x-axis (y < 0).Always sketch the parabola — even a rough U-shape takes 5 seconds and prevents this error completely.The sketch tells you whether you want inside the roots or outside the roots.

Diagnosticrecall

Solve x² − 16 < 0

1 mark2 minsquadratic-inequalities-q1

Show solution

Worked solution

1.Spot the skill: Sketch the parabola: find the roots, then decide which region satisfies the inequality.
2.Use the factorise the quadratic to find the roots stage first, then sketch the parabola.
3.Keep the final answer visible: −4 < x < 4.

Final answer

−4 < x < 4

Mark points

M1: use the correct sketch the parabola: find the roots, then decide which region satisfies the inequality.for ax² > 0 (positive leading coefficient), the parabola opens upward — it is above the x-axis outside the roots.for ax² < 0, it is below the x-axis between the roots.
A1: −4 < x < 4

Watch out

Students write −3 < x < 2 (the region BETWEEN the roots) when the inequality is > 0 and the parabola opens upward.That region is actually below the x-axis (y < 0).Always sketch the parabola — even a rough U-shape takes 5 seconds and prevents this error completely.The sketch tells you whether you want inside the roots or outside the roots.

Easyprocedure

Solve x² − 5x + 4 > 0

2 marks3 minsquadratic-inequalities-q2

Show solution

Worked solution

1.Spot the skill: Sketch the parabola: find the roots, then decide which region satisfies the inequality.
2.Use the sketch the parabola stage first, then identify the region where the graph is above the x-axis (y > 0).
3.Keep the final answer visible: x < 1 or x > 4.

Final answer

x < 1 or x > 4

Mark points

M1: use the correct sketch the parabola: find the roots, then decide which region satisfies the inequality.for ax² > 0 (positive leading coefficient), the parabola opens upward — it is above the x-axis outside the roots.for ax² < 0, it is below the x-axis between the roots.
A1: x < 1 or x > 4

Watch out

Students write −3 < x < 2 (the region BETWEEN the roots) when the inequality is > 0 and the parabola opens upward.That region is actually below the x-axis (y < 0).Always sketch the parabola — even a rough U-shape takes 5 seconds and prevents this error completely.The sketch tells you whether you want inside the roots or outside the roots.

Mediumreasoning

Solve x² − 3x − 10 ≤ 0

3 marks4 minsquadratic-inequalities-q3

Show solution

Worked solution

1.Spot the skill: Sketch the parabola: find the roots, then decide which region satisfies the inequality.
2.Use the identify the region where the graph is above the x-axis (y > 0) stage first, then write the solution.
3.Keep the final answer visible: −2 ≤ x ≤ 5.

Final answer

−2 ≤ x ≤ 5

Mark points

M1: use the correct sketch the parabola: find the roots, then decide which region satisfies the inequality.for ax² > 0 (positive leading coefficient), the parabola opens upward — it is above the x-axis outside the roots.for ax² < 0, it is below the x-axis between the roots.
A1: −2 ≤ x ≤ 5

Watch out

Students write −3 < x < 2 (the region BETWEEN the roots) when the inequality is > 0 and the parabola opens upward.That region is actually below the x-axis (y < 0).Always sketch the parabola — even a rough U-shape takes 5 seconds and prevents this error completely.The sketch tells you whether you want inside the roots or outside the roots.

Hardproblem solving

Solve 2x² − 5x − 3 > 0

3 marks5 minsquadratic-inequalities-q4

Show solution

Worked solution

1.Spot the skill: Sketch the parabola: find the roots, then decide which region satisfies the inequality.
2.Use the write the solution stage first, then factorise the quadratic to find the roots.
3.Keep the final answer visible: x < − $\frac{1}{2}$ or x > 3.

Final answer

x < − $\frac{1}{2}$ or x > 3

Mark points

M1: use the correct sketch the parabola: find the roots, then decide which region satisfies the inequality.for ax² > 0 (positive leading coefficient), the parabola opens upward — it is above the x-axis outside the roots.for ax² < 0, it is below the x-axis between the roots.
A1: x < − $\frac{1}{2}$ or x > 3

Watch out

Students write −3 < x < 2 (the region BETWEEN the roots) when the inequality is > 0 and the parabola opens upward.That region is actually below the x-axis (y < 0).Always sketch the parabola — even a rough U-shape takes 5 seconds and prevents this error completely.The sketch tells you whether you want inside the roots or outside the roots.

Exam-stylemulti-step

Find the set of integer values satisfying x² − 6x + 5 < 0

4 marks6 minsquadratic-inequalities-q5

Show solution

Worked solution

1.Spot the skill: Sketch the parabola: find the roots, then decide which region satisfies the inequality.
2.Use the factorise the quadratic to find the roots stage first, then sketch the parabola.
3.Keep the final answer visible: {2, 3, 4}.

Final answer

{2, 3, 4}

Mark points

M1: use the correct sketch the parabola: find the roots, then decide which region satisfies the inequality.for ax² > 0 (positive leading coefficient), the parabola opens upward — it is above the x-axis outside the roots.for ax² < 0, it is below the x-axis between the roots.
A1: {2, 3, 4}

Watch out

Students write −3 < x < 2 (the region BETWEEN the roots) when the inequality is > 0 and the parabola opens upward.That region is actually below the x-axis (y < 0).Always sketch the parabola — even a rough U-shape takes 5 seconds and prevents this error completely.The sketch tells you whether you want inside the roots or outside the roots.

Grade 9 stretchproblem solving

Solve x² - 2x - 15 ≥ 0.

4 marks7 minsquad-ineq-g9

Show solution

Worked solution

1.Factorise the quadratic.
2.Find its roots.
3.Use the regions outside the roots because the parabola opens upwards.

Final answer

x ≤ -3 or x ≥ 5

Mark points

M1: (x - 5)(x + 3)
M1: roots -3 and 5
A1: correct regions

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

16 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Quadratic inequalities - 2 marksSolve x² − 16 < 0Mark answer

Answer

−4 < x < 4

2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer

Answer

−4

3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer

Answer

3x²y + xy

4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer

Answer

3x(2x + 3)

Use the graph to choose inside or outside the roots

What you will learn

Know the rule, then use it

Method

Factorise the quadratic to find the roots

Sketch the parabola

Identify the region where the graph is ABOVE the x-axis (y > 0)

Watch out

Solve x² + x − 6 > 0

Build up to the hardest questions

Switch between skills

Get help with anything that still feels tricky.