Here is a student's solution to an inequality. -4x + 7 > 23 -4x > 16 x > -4 Which statement about this working is correct?

The last line should be x < -4, because both sides were divided by a negative number.

The second line should be -4x > 30.

Are quadratic inequalities on the Foundation paper?

No. Foundation tier covers A22 only as far as linear inequalities in one variable and showing the solution on a number line. Quadratic inequalities and inequalities in two variables are Higher tier only, which is why they tend to appear late in the paper where the grade 7-9 marks sit.

How many marks are inequalities worth on Edexcel Higher?

A single linear inequality is usually 2-3 marks, often with a follow-up part worth 1-2 more for a number line or the integer solutions. A quadratic inequality is typically 3 marks, rising to 5 when it is combined with a linear inequality and you have to give the overlap of the two solution sets.

Why does the inequality sign flip when you divide by a negative?

Because multiplying by a negative reverses the order of the number line. 2 -5. The same reversal happens whenever you multiply or divide an inequality by a negative quantity, so the sign has to turn round with it.

Do I have to write the answer in set notation?

Only when the question asks for it. Edexcel lists set notation as one of the accepted ways of representing a solution set alongside a number line and a graph, so a question can specify it; a plain inequality such as x > 3 is accepted otherwise. Set notation is written with curly brackets and a colon, and two separate ranges are joined with the union symbol.

Inequalities - Edexcel GCSE Higher Maths

Typically 3-6 marks per paper, more in a series where a quadratic inequality appears · Spec 1MA1

Spec point A22 asks you to solve linear inequalities in one or two variables and quadratic inequalities in one variable, then show the solution set on a number line, in set notation, or as a region on a graph. Foundation candidates meet only the one-variable linear case, shown on a number line. The two-variable regions, the quadratic work and set notation all belong to Higher, and a quadratic inequality generally sits in the grade 7-9 stretch near the end of a paper.

Edexcel spreads the topic across all three papers. A short 'solve the inequality' item worth 2-3 marks turns up in the first half, often followed by a part asking for the largest integer that satisfies it or for a number line to be completed. The quadratic version is worth 3-5 marks and can be paired with a linear inequality, so the answer is the overlap of two solution sets rather than two separate ranges. Paper 1 is non-calculator, so a quadratic inequality set there will factorise; on Papers 2 and 3 a calculator is allowed, so the critical values can be messier.

Two slips account for most of the marks lost. The first is dividing by a negative and leaving the sign where it was, turning $-3x > 12$ into $x > -4$ when the answer is $x < -4$ . The second is swapping the inequality sign for an equals sign partway down the working and never putting it back, which collapses a solution set into a single value and loses the accuracy mark even when the arithmetic is perfect. Practise answering the question as it is worded: 'list the integers' is not satisfied by a range, and 'solve' is not satisfied by a list.

Worked example

Find algebraically the set of values of

x

for which

x^2 - x - 20 < 0

and

4x + 1 > 9

[5 marks]

Factorise the quadratic: $x^2 - x - 20 = (x - 5)(x + 4)$ .
M1: a correct factorisation (or correct use of the quadratic formula) leading to the critical values
The critical values are $x = -4$ and $x = 5$ . The curve $y = x^2 - x - 20$ is U-shaped, so it lies below the $x$ -axis between the roots: $-4 < x < 5$ .
A1: both critical values. A1: the region between them, not the two outside pieces
Solve the linear inequality on its own: $4x + 1 > 9$ gives $4x > 8$ , so $x > 2$ .
B1: correct solution of the linear inequality, sign kept throughout
Both conditions have to hold at once. Sketch the two ranges one above the other on a number line: $-4 < x < 5$ and $x > 2$ overlap where $2 < x < 5$ .
A1: the overlap as a single interval; two separate ranges joined by 'and' scores nothing here

Practice questions

These are original questions in Edexcel 1MA1 Higher style, covering linear inequalities, unknowns on both sides, double inequalities, integer solution sets, regions and the Higher-only quadratic inequalities. Number lines cannot be drawn here, so those questions describe the line in words and ask you to pick the matching inequality. Try the quadratic ones without a calculator first, the way Paper 1 would set them.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

Question 1 Warm-up [1 mark]

An inequality in $x$ is shown on a number line by an open (unfilled) circle at 2, with an arrow from the circle pointing to the right. Which inequality is shown?

Answer options for question 1
$x > 2$ $x ≥ 2$ $x < 2$ $x ≤ 2$
Question 2 Warm-up [2 marks]

$n$ is an integer such that $-3 ≤ n < 2$ . Work out how many values $n$ can take.

Your answer (fractions like 1/2 are fine)
Question 3 Exam pace [3 marks]

Solve the inequality $3(x - 4) > 5x + 2$ . Write down the largest integer value of $x$ that satisfies it.

Your answer (fractions like 1/2 are fine)
Question 4 Exam pace [3 marks]

Solve the inequality $2(3x + 1) ≤ 5(x - 2)$ . Write down the largest integer value of $x$ that satisfies it.

Your answer (fractions like 1/2 are fine)
Question 5 Exam pace [2 marks]

Here is a student's solution to an inequality. $-4x + 7 > 23$ $-4x > 16$ $x > -4$ Which statement about this working is correct?

Answer options for question 5
The last line should be $x < -4$ , because both sides were divided by a negative number. The last line should be $x > 4$ . The second line should be $-4x > 30$ . The working is correct.
Question 6 Stretch [3 marks]

Work out how many integer values of $x$ satisfy $x^2 - 2x - 8 < 0$ .

Your answer (fractions like 1/2 are fine)

Common mistakes examiners see

Dividing or multiplying both sides by a negative number and leaving the sign alone, so $-3x > 12$ is answered as $x > -4$ instead of $x < -4$ .

Move the $x$ term to whichever side makes it positive before you divide. Adding $3x$ to both sides of $-3x > 12$ gives $0 > 12 + 3x$ , then $-12 > 3x$ and $x < -4$ , with no negative division anywhere in the working.
Rewriting the inequality as an equation, solving it, and giving a single value such as $x = 4$ as the answer.

Carry the sign down every line of working. An answer with an equals sign in it cannot score the accuracy mark on an inequality question, however good the arithmetic that produced it.
Giving the outside region of a quadratic when the inside is wanted, for example answering $x^2 - 3x - 10 < 0$ with $x < -2$ or $x > 5$ .

Sketch the parabola, mark the two roots on the $x$ -axis, and decide which part of the curve is below the axis (for $< 0$ ) or above it (for $> 0$ ). A rough sketch takes seconds and settles the direction every time.
Operating on only part of a double inequality, so $-7 < 2x + 3 ≤ 11$ becomes $-7 < 2x ≤ 8$ with the left-hand end untouched.

Do the same thing to all three parts on the same line, and check afterwards that the two ends still bracket a sensible range.
Using the wrong endpoint convention: a solid circle for a strict inequality on a number line, or including an endpoint in a list, such as writing $-2, -1, 0, 1, 2, 3$ for the integers satisfying $-2 < n ≤ 3$ .

Fill the circle and include the endpoint only when the sign carries the bar ( $≤$ or $≥$ ). Read your final line back against the original inequality before you write the list.

Frequently asked questions

Are quadratic inequalities on the Foundation paper?: No. Foundation tier covers A22 only as far as linear inequalities in one variable and showing the solution on a number line. Quadratic inequalities and inequalities in two variables are Higher tier only, which is why they tend to appear late in the paper where the grade 7-9 marks sit.
How many marks are inequalities worth on Edexcel Higher?: A single linear inequality is usually 2-3 marks, often with a follow-up part worth 1-2 more for a number line or the integer solutions. A quadratic inequality is typically 3 marks, rising to 5 when it is combined with a linear inequality and you have to give the overlap of the two solution sets.
Why does the inequality sign flip when you divide by a negative?: Because multiplying by a negative reverses the order of the number line. $2 < 5$ is true, but multiplying both sides by $-1$ gives $-2$ and $-5$ , and $-2 > -5$ . The same reversal happens whenever you multiply or divide an inequality by a negative quantity, so the sign has to turn round with it.
Do I have to write the answer in set notation?: Only when the question asks for it. Edexcel lists set notation as one of the accepted ways of representing a solution set alongside a number line and a graph, so a question can specify it; a plain inequality such as $x > 3$ is accepted otherwise. Set notation is written with curly brackets and a colon, and two separate ranges are joined with the union symbol.

Inequalities - Edexcel GCSE Higher Maths

Worked example

Practice questions

Common mistakes examiners see

Frequently asked questions

Related topics