Quick guide. How to solve a quadratic inequality, combine it with a linear inequality on a number line, examiner tips and four exam-style questions with hidden solutions.
This page is for AS Maths and GCSE Higher students revising quadratic and linear inequalities and for maths teachers who need a quick, ready-made lesson. Watch the video, skim the diagrams, then try the exam-style questions. Teachers can project each question full screen for the whole class.
Video Tutorial: Solving Quadratic and Linear Inequalities
In the video:
▶️ Recap: solving a single quadratic inequality from a sketched graph
▶️ Sketching a quadratic and a linear graph on the same axes
▶️ Finding the points of intersection algebraically
▶️ Challenge: solving a pair of quadratic inequalities
👩🏫 Teaching this topic? The full Quadratic and Linear Inequalities lesson pack includes the editable PowerPoint, differentiated worksheet and worked solutions, ready to teach in one download.
What is a Quadratic Inequality?
A quadratic inequality asks where a quadratic expression is positive or negative, rather than where it equals zero. The sketch does the work:
Find the roots. Factorise or use the quadratic formula
Sketch the parabola. Positive x² coefficient opens upwards
Read the region. Above the x-axis for > 0, below for < 0
x² − 2x − 3 > 0 when x < −1 or x > 3 (curve above the axis). x² − 2x − 3 < 0 when −1 < x < 3 (curve below the axis).
💡 For an upward parabola: > 0 means outside the roots (two separate parts joined by “or”) and < 0 means between the roots (one interval).
How to Solve Quadratic and Linear Inequalities Together
Exam questions often ask for the values of x that satisfy a quadratic inequality and a linear inequality at the same time. The method:
Solve the quadratic inequality with a sketched parabola
Solve the linear inequality by rearranging (reverse the sign if you divide by a negative)
Combine on a number line. The answer is the overlap, the values in both solution sets
When the question compares two graphs
Some questions phrase the same idea graphically: for which values of x is the curve f(x) above or below the line g(x)? Solve f(x) = g(x) to find the intersections, then read the regions from the sketch.
The graphs cross where f(x) = g(x), here at x = 2 and x = 5.
f(x) > g(x) where the parabola is above the line: x < 2 or x > 5
f(x) < g(x) where the parabola is below the line: 2 < x < 5
The solutions to f(x) = g(x) are x = 2 and x = 5
Worked Example
Solve x² − 4x − 5 ≤ 0 and 2x − 3 ≥ 1.
Solve the quadratic. x² − 4x − 5 = (x − 5)(x + 1) ≤ 0. The parabola is below the axis between the roots: −1 ≤ x ≤ 5
Solve the linear. 2x − 3 ≥ 1 so 2x ≥ 4, giving x ≥ 2
Combine on a number line. The overlap of the two solution sets is the answer
Only the values on both bars satisfy both inequalities.
2 ≤ x ≤ 5
💡 Sense check: pick a value in the answer, say x = 3. Then 9 − 12 − 5 = −8 ≤ 0 ✅ and 6 − 3 = 3 ≥ 1 ✅
Examiner Tips and Tricks
🎯 What examiners look for
Always sketch the parabola. Students who skip the sketch often give the complement of the correct region.
“And” means overlap. Draw both solution sets on one number line and take the values common to both.
Track strict and inclusive signs separately. If the quadratic gives 2 ≤ x ≤ 3 but the linear gives x > 2, the combined answer is 2 < x ≤ 3. The endpoint rules come from each inequality individually.
Dividing by a negative reverses the sign. 5 − 2x ≤ 1 gives −2x ≤ −4, so x ≥ 2, not x ≤ 2.
Integer solutions come last. Solve the inequality fully first, then list the integers inside the final interval, checking whether each endpoint is included.
Comparing graphs? Solve f(x) = g(x) for the intersections, then read whether the question wants the parabola above or below the line.
Exam-Style Questions
Teachers: use the full screen button to project each question for the whole class. Students: try each question on paper before opening the solution.
Question 1
Solve for x:
x² − 5x + 6 ≤ 0 and 2x − 1 > 3
Write down the integer values of x that satisfy both inequalities.
Show worked solution
Quadratic: x² − 5x + 6 = (x − 2)(x − 3) ≤ 0. Below the axis between the roots: 2 ≤ x ≤ 3
Linear: 2x − 1 > 3 so 2x > 4, giving x > 2
Combine: both hold when 2 < x ≤ 3. The value x = 2 is excluded because the linear inequality is strict
Integers: the only integer in 2 < x ≤ 3 is 3
2 < x ≤ 3, integer solution: {3}
Question 2
Solve for x:
x² − 2x − 8 < 0 and 3x + 1 ≥ 10
Write down the integer values of x that satisfy both inequalities.
Show worked solution
Quadratic: x² − 2x − 8 = (x − 4)(x + 2) < 0. Below the axis between the roots: −2 < x < 4
Linear: 3x + 1 ≥ 10 so 3x ≥ 9, giving x ≥ 3
Combine: both hold when 3 ≤ x < 4. The value x = 3 is included but x = 4 is not
Integers: the only integer in 3 ≤ x < 4 is 3
3 ≤ x < 4, integer solution: {3}
Question 3
Solve for x:
x² − 7x + 12 ≤ 0 and 5 − 2x ≤ 1
Write down the integer values of x that satisfy both inequalities.
Show worked solution
Quadratic: x² − 7x + 12 = (x − 3)(x − 4) ≤ 0. Below the axis between the roots: 3 ≤ x ≤ 4
Linear: 5 − 2x ≤ 1 so −2x ≤ −4. Dividing by −2 reverses the sign: x ≥ 2
Combine: the quadratic interval already lies inside x ≥ 2, so both hold when 3 ≤ x ≤ 4
Integers: the integers in 3 ≤ x ≤ 4 are 3 and 4
3 ≤ x ≤ 4, integer solutions: {3, 4}
Question 4
Solve for x:
x² − x − 12 < 0 and 2x + 3 > 5
Write down the integer values of x that satisfy both inequalities.
Show worked solution
Quadratic: x² − x − 12 = (x − 4)(x + 3) < 0. Below the axis between the roots: −3 < x < 4
Linear: 2x + 3 > 5 so 2x > 2, giving x > 1
Combine: both hold when 1 < x < 4. Both endpoints are excluded because both inequalities are strict
Integers: the integers in 1 < x < 4 are 2 and 3
1 < x < 4, integer solutions: {2, 3}
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