Solve inequalities such as x² − 7x + 10 > 0 by finding the critical values and reading the correct region from a sketch. It is Higher only, aimed at grade 8. Free questions with answers are on this page.
Solve the corresponding equation first, sketch the parabola (or a number line with the two roots), and read where the curve is above or below the axis. x2−5x+6<0 is between the roots: 2<x<3. x2−5x+6≥0 is outside: x≤2 or x≥3. A filled circle for ≤; empty for <.
If a<0 the U is upside-down, so the 'above the axis' region flips. Do not write 2<x>3. Factorising, formula or graph — any legal way to find the critical values is fine. Integer solutions can be listed if asked.
Factorise (or complete the square), find the critical values, sketch the U-shaped graph. <0 is between the roots; >0 is outside. Open circles for strict inequalities; include the ends for ≤ and ≥. Then list integers if asked.
Objectives
Find both roots.
Choose inside or outside from the inequality sign.
Keep the inequality strict or inclusive as written.
Key points
Rearrange so one side is 0 and the x2 term is positive.
Factorise (or use the formula) to find the critical values where the expression equals 0.
Sketch the U-shaped curve and decide whether you want the part above or below the x-axis.
Write the solution correctly: one combined inequality for 'between', two separate inequalities joined by 'or' for 'outside'.
Worked examples
Example 1
Solve x^2-5x+6<0.
2 marks
Roots 2 and 3
The parabola is negative between the roots
2<x<3
Answer: 2<x<3
Common slips
Slip: Root 6 only Factor (x-2)(x-3).
Slip: It is negative outside a is positive, so the dip is in the middle.
Slip: x<2 or x>3 The outside region belongs to the greater-than inequality.
Example 2
Solve x^2-5x+6>0.
2 marks
Same roots 2 and 3
Positive outside the roots
x<2 or x>3
Answer: x<2 or x>3
Common slips
Slip: Roots change when the sign changes The boundary is the same equation.
Slip: Positive between the roots The U-shape is above the axis outside.
Slip: 2<x<3 Join the two outer parts with or.
Example 3
Solve 2x^2-8x+6<0.
3 marks
Divide by the positive 2: x^2-4x+3<0
Roots 1 and 3
1<x<3
Answer: 1<x<3
Common slips
Slip: The inequality flips when you divide by 2 2 is positive, so the sign stays.
Slip: Roots 2 and 4 (x-1)(x-3)=0.
Slip: x<1 or x>3 Less than means between the roots.
Questions and answers
GCSE Maths quadratic inequalities questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Give the solution set of (x-2)(x-3)<0.
Show answer
Answer: 2<x<3
The roots are 2 and 3.
The product is negative when one bracket is positive and the other is negative, which happens between 2 and 3.
M1 — The roots are 2 and 3.
A1 — Answer 2<x<3
Question 2 Easy2 marks
Solve x^2-4<0.
Show answer
Answer: -2<x<2
Roots -2 and 2.
The square is below 4 between them.
M1 — Roots -2 and 2.
A1 — Answer -2<x<2
Question 3 Medium2 marks
Solve x^2-5x+4<=0.
Show answer
Answer: 1<=x<=4
Roots 1 and 4.
Inclusive, so keep both ends.
M1 — Roots 1 and 4.
A1 — Answer 1<=x<=4
Question 4 Medium2 marks
Solve x^2-5x+4>=0.
Show answer
Answer: x<=1 or x>=4
Roots 1 and 4.
Inclusive and outside.
M1 — Roots 1 and 4.
A1 — Answer x<=1 or x>=4
Question 5 Hard2 marks
Solve 2x^2-8x+6>0.
Show answer
Answer: x<1 or x>3
Divide by positive 2 first.
Outside the roots 1 and 3.
M1 — Divide by positive 2 first.
A1 — Answer x<1 or x>3
Question 6 Hard2 marks
A quadratic graph with a positive x^2 is below the x-axis only between x=2 and x=3. Write this as an inequality in x.
Show answer
Answer: 2<x<3
Below means less than.
At 2 and 3 the graph is on the axis, not below it, so the signs stay strict.
M1 — Below means less than.
A1 — Answer 2<x<3
Question 7 Exam-style5 marks
Solve 2x^2-2x-12>=0.
Show answer
Answer: x<=-2 or x>=3
Divide by 2.
x^2-x-6>=0.
Roots -2 and 3.
Outside, inclusive.
x<=-2 or x>=3.
M1 — Divide by 2
M1 — Factor
A1 — Roots
M1 — Outside
A1 — Inclusive region
Common mistakes
Most lost marks on this topic are one slip.
Writing the outside region for a less-than sign
With a positive x^2, less than means between the roots and greater than means outside them
Flipping the sign when dividing by a positive number
Only dividing or multiplying by a negative number flips the sign
Leaving out the roots for <= or >=
An inclusive sign keeps both roots in the answer
Spot the slip
The roots of x^2-5x+6=0 are 2 and 3. A student gives the solution of x^2-5x+6<0 as x<2 or x>3. What region did they swap?
They described where the graph is above the axis
2 and 3 are not the roots
A quadratic has only one region
Show the slip
They described where the graph is above the axis With a positive x^2, the graph is below the axis between the roots.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
A tiny sketch (U-shape, two crossings labelled) is quick and usually secures the method mark even if you misread the region.
Always finish with the solution in inequality form — a shaded number line alone is not a full answer unless the question asks for one.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.