What is A22: Inequalities?

Solve a linear inequality like an equation, reversing the inequality sign only when multiplying or dividing by a negative number.

How many marks are A22 questions worth?

In verified past-paper appearances, A22 questions on Edexcel GCSE Maths 1MA1 have been worth 2–6 marks.

Which past exam papers has A22 appeared in?

A22 has appeared in 25 verified past-paper questions indexed on this page, with direct links to the official question paper and mark scheme for each.

A22

Solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph

Inequalities

Worked answers, methods and verified real exam appearances for A22 on Edexcel GCSE Maths 1MA1.

Explanation

Solve a linear inequality like an equation, reversing the inequality sign only when multiplying or dividing by a negative number.
On a number line, use a filled endpoint when equality is included and an open endpoint for a strict inequality.
At Higher tier, represent two-variable inequalities by drawing each boundary and testing a point to choose the region; strict boundaries are dashed.
For a quadratic inequality, find the roots and test the intervals they define, because the required values may lie inside or outside the roots.
The examiner expects the correct endpoint style, shading and notation as well as the algebraic boundary values.

A strict inequality outside two critical values uses open endpoints and outward shading.

Worked example

Higher tier: Solve $(2x+3)(x-2)>0$ .

1.Find the critical values: $2x+3=0$ gives $x=-\dfrac32$ , and $x-2=0$ gives $x=2$ .
2.Test the three intervals; the product is positive outside the roots.
3.The inequality is strict, so neither endpoint is included.

Answer: $x<-\dfrac32$ or $x>2$ .

Common mistakes

Don't forget to reverse the inequality sign after dividing by a negative number.
Don't use filled endpoints for $<$ or $>$ .
Don't assume a quadratic inequality is always satisfied between its roots without testing signs.

Exam tip

For a graphical answer, the boundary style and the direction of shading are separate marking points.

Worked practice

Tier 1 · Easy

Higher only: Solve

3x-7\le11

. Give the answer in set notation and describe its number-line representation.

(3)

(Total for Question 1 is 3 marks)

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Mark scheme
1	$\{x:x\le6\}$ A filled point at $6$ with the line shaded to the left.	3	Add $7$ to get $3x\le18$ , then divide by $3$ to obtain $x\le6$ . Equality is included, so use a filled point at $6$ and shade all smaller values.

Tier 2 · Standard

Higher only: On coordinate axes, show the region satisfying both

y\ge2x-1

and

x+y<5

. State the intersection of the boundary lines and identify which boundaries are included.

(5)

(Total for Question 2 is 5 marks)

Mark scheme

Mark scheme for question 2
Question	Answer	Mark	Mark scheme
2	Shade above $y=2x-1$ and below $y=5-x$ The boundaries meet at $(2,3)$ $y=2x-1$ is solid and $y=5-x$ is dashed	5	Draw $y=2x-1$ as a solid line because equality is allowed. Draw $y=5-x$ as a dashed line because $x+y<5$ is strict. Solving $2x-1=5-x$ gives $3x=6$ , so the lines meet at $(2,3)$ . The required region is above the solid line and below the dashed line.

Tier 3 · Hard

Higher only: Solve

(2x+3)(x-2)>0

. Give the solution in set notation and describe it on a number line.

(4)

(Total for Question 3 is 4 marks)

Mark scheme

Mark scheme for question 3
Question	Answer	Mark	Mark scheme
3	$\{x:x<-\frac32\text{ or }x>2\}$ Open points at $-\frac32$ and $2$ , shaded outwards.	4	The critical values are $x=-3/2$ and $x=2$ . The product is positive outside these roots, so $x<-3/2$ or $x>2$ . The inequality is strict, so both endpoints are open and the two outer regions are shaded.

Tier 1 · Easy

Solve

-3x>12

and describe the solution on a number line.

(2)

(Total for Question 4 is 2 marks)

Mark scheme

Mark scheme for question 4
Question	Answer	Mark	Mark scheme
4	$x<-4$ An open point at $-4$ with the line shaded to the left.	2	Dividing both sides by $-3$ reverses the inequality, giving $x<-4$ . The inequality is strict, so use an open point at $-4$ and shade to the left.

Tier 2 · Standard

Solve

5\le2x+1\le13

and represent the solution on a number line.

(3)

(Total for Question 5 is 3 marks)

Mark scheme

Mark scheme for question 5
Question	Answer	Mark	Mark scheme
5	$2\le x\le6$ Filled points at $2$ and $6$ , with the line shaded between them.	3	Subtract $1$ throughout to get $4\le2x\le12$ . Divide throughout by $2$ , giving $2\le x\le6$ . Both endpoints are included, so use filled points and shade between them.

Tier 3 · Hard

Solve

3x+2>11

and

5x-4\le31

. Hence write down all the positive integer values of

x

that satisfy both inequalities.

(3)

(Total for Question 6 is 3 marks)

Mark scheme

Mark scheme for question 6
Question	Answer	Mark	Mark scheme
6	$3<x\le7$ $x=4,5,6,7$	3	The first inequality gives $3x>9$ , so $x>3$ . The second gives $5x\le35$ , so $x\le7$ . Their intersection is $3<x\le7$ , and the positive integers in this interval are $4,5,6,7$ .

Tier 2 · Standard

A shelf can support at most

85

kg. Five identical boxes and an item of mass

12

kg are put on the shelf. Form and solve an inequality for the mass

m

kg of each box. Hence write down the greatest possible whole-number mass of one box.

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
Question	Answer	Mark	Mark scheme
7	$m\le14.6$ The greatest possible whole-number mass is $14$ kg.	3	The mass condition is $5m+12\le85$ . Therefore $5m\le73$ , so $m\le14.6$ . The greatest whole number satisfying this is $14$ .

Tier 3 · Hard

Higher only: On coordinate axes, show the region satisfying

x\ge1

y\ge0

y\le x+2

and

2x+y<11

. Work out the intersection of the boundary lines

y=x+2

and

2x+y=11

. Identify which of the four boundaries are included in the region.

(5)

(Total for Question 8 is 5 marks)

Mark scheme

Mark scheme for question 8
Question	Answer	Mark	Mark scheme
8	Shade the region to the right of $x=1$ , above $y=0$ , below $y=x+2$ and below $2x+y=11$ The two sloping boundaries meet at $(3,5)$ $x=1$ , $y=0$ and $y=x+2$ are included; $2x+y=11$ is not included	5	Draw $x=1$ , $y=0$ and $y=x+2$ as solid lines because equality is included. Draw $2x+y=11$ as a dashed line because the inequality is strict, then shade the common region. For the sloping-line intersection, substitute $y=x+2$ into $2x+y=11$ : $3x+2=11$ , so $x=3$ and $y=5$ .

Tier 3 · Hard

Higher only: Solve

x^2+x-20\ge0

and give the solution in set notation. Hence write down all the integer solutions that also satisfy

-7<x<6

(5)

(Total for Question 9 is 5 marks)

Mark scheme

Mark scheme for question 9
Question	Answer	Mark	Mark scheme
9	$\{x:x\le-5\text{ or }x\ge4\}$ $x=-6,-5,4,5$	5	Factorise to get $(x+5)(x-4)\ge0$ . The product is non-negative outside the roots, including both endpoints, so $x\le-5$ or $x\ge4$ . Intersecting this set with $-7<x<6$ leaves the integers $-6,-5,4,5$ .

Q10

Tier 3 · Hard

Higher only: Solve the compound inequality

\dfrac{3x-2}{4}<\dfrac{x+5}{3}\le\dfrac{2x+7}{2}

. Give the solution in set notation and write down all the positive integer values of

x

in the solution set.

(5)

(Total for Question 10 is 5 marks)

Mark scheme

Mark scheme for question 10
Question	Answer	Mark	Mark scheme
10	$\left\{x:-\dfrac{11}{4}\le x<\dfrac{26}{5}\right\}$ $x=1,2,3,4,5$	5	The left inequality gives $3(3x-2)<4(x+5)$ , so $9x-6<4x+20$ and $x<26/5$ . The right inequality gives $2(x+5)\le3(2x+7)$ , so $2x+10\le6x+21$ and $x\ge-11/4$ . Combining the bounds gives $-11/4\le x<26/5$ , whose positive integer values are $1,2,3,4,5$ .

Verified exam appearances

Series	Paper	Question	Marks	Calculator	Tier	Links
2024-06	2H	Q16	2	Allowed	Higher	QP MS
2021-11	1H	Q17	3	Non-calculator	Higher	QP MS
2023-11	1H	Q17	4	Non-calculator	Higher	QP MS
2024-11	2F	Q19	4	Allowed	Foundation	QP MS
2023-06	1H	Q24	5	Non-calculator	Higher	QP MS
2024-06	2H	Q13	3	Allowed	Higher	QP MS
2023-06	2F	Q23	6	Allowed	Foundation	QP MS
2022-11	3H	Q11	2	Allowed	Higher	QP MS
2022-11	1F	Q26	5	Non-calculator	Foundation	QP MS
2019-11	2H	Q23	6	Allowed	Higher	QP MS
2021-11	2H	Q1	3	Allowed	Higher	QP MS
2022-11	2H	Q16	4	Allowed	Higher	QP MS
2022-06	1H	Q1	2	Non-calculator	Higher	QP MS
2023-11	3F	Q23	4	Allowed	Foundation	QP MS
2021-11	2F	Q21	3	Allowed	Foundation	QP MS
2019-11	1F	Q28	3	Non-calculator	Foundation	QP MS
2024-06	1F	Q28	3	Non-calculator	Foundation	QP MS
2023-06	2H	Q4	4	Allowed	Higher	QP MS
2023-11	3H	Q4	4	Allowed	Higher	QP MS
2019-11	1F	Q19	2	Non-calculator	Foundation	QP MS
2019-06	2H	Q1	5	Allowed	Higher	QP MS
2019-06	2F	Q20	5	Allowed	Foundation	QP MS
2019-06	3H	Q18	6	Allowed	Higher	QP MS
2022-06	2H	Q22	5	Allowed	Higher	QP MS
2022-06	1F	Q23	2	Non-calculator	Foundation	QP MS

Other points in A Algebra · sequences and functions

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