Inequalities - Edexcel GCSE Maths (Calculator) Exam Questions

Practise 29 exam-style questions on Inequalities for Edexcel GCSE Maths (Calculator). Write your answer as you would in the exam and get it marked in seconds against the mark scheme, point by point, with an AI tutor on hand if you get stuck.

29 questions
1–5 marks each
Instant marking
No sign-up needed

20 questions, marked as you go

Sample questions with model answers

Here are 4 of the questions in this topic, each with a model answer, an explanation of why it earns the marks and the mark scheme an examiner would use.

Question 1

2 marks

\(n\) is an integer such that \(-7.5 < 3n \le 9.6\)

Write down all the possible values of \(n\).

Model answer

Divide all three parts by 3:
\(-2.5 < n \le 3.2\)
\(n = -2, -1, 0, 1, 2, 3\)

Explanation

Divide every part of the double inequality by 3 to get \(-2.5 < n \le 3.2\). The integers in this range start at \(-2\) (since \(-3\) is less than \(-2.5\)) and end at 3. Remember to include 0, and do not include \(-3\) or 4.

Mark scheme

1 mark
B1 for \(-2.5 < n \le 3.2\), or for a list with one error or omission
exact values
1 mark
B1 for \(-2, -1, 0, 1, 2, 3\)
cao; no extra values; B2 for a fully correct list with no working

Question 2

2 marks

The number line shows the solution set of the inequality
\(a < 2x + 1 \le b\)
where \(a\) and \(b\) are integers.

Find the value of \(a\) and the value of \(b\).

A number line from −5 to 3, with a mark at every 0.5. Above it is an open (hollow) circle on the left joined by a line to a filled (solid) circle on the right. The positions of the circles can be read from the number line.

Model answer

The number line shows \(-3 < x \le 1.5\)
Multiply by 2: \(-6 < 2x \le 3\)
Add 1: \(-5 < 2x + 1 \le 4\)
\(a = -5\), \(b = 4\)

Explanation

Read the inequality from the number line: an open circle at \(-3\) and a filled circle at 1.5 give \(-3 < x \le 1.5\). Then build up \(2x + 1\) by doing the same to all three parts: multiply by 2, then add 1.

Mark scheme

1 mark
B1 for \(a = -5\)
cao; correct answer with no working scores full marks
1 mark
B1 for \(b = 4\)
cao

Question 3

3 marks

A school has £12 000 to spend on laptops and tablets.
A laptop costs £385 and a tablet costs £249
The school buys \(x\) laptops and \(y\) tablets, and must not spend more than £12 000

The school decides to buy 18 tablets.
Work out the greatest number of laptops it can buy.

Model answer

\(385x + 249y \le 12000\) with \(y = 18\):
\(385x + 4482 \le 12000\)
\(385x \le 7518\)
\(x \le 19.52...\)
The greatest number of laptops is 19.

Explanation

The total cost \(385x + 249y\) must be at most £12 000, which gives a linear inequality in two variables. Substituting \(y = 18\) leaves an inequality in \(x\). The number of laptops must be a whole number, so round 19.52... down to 19, because 20 laptops would cost too much.

Mark scheme

1 mark
M1 for \(385x + 249 \times 18 \le 12000\) or for \(12000 - 249 \times 18\) (= 7518)
oe; correct answer with no working scores full marks
1 mark
M1 for \(x \le \dfrac{7518}{385}\) or for 19.5...
oe
1 mark
A1 for 19
cao; do not accept 20

Question 4

5 marks

A ball is kicked upwards from a height of 1.5 m.
The height, \(h\) metres, of the ball above the ground \(t\) seconds after it is kicked is given by
\(h = 1.5 + 22t - 4.9t^2\)

For how many seconds is the ball more than 20 m above the ground?
Give your answer correct to 2 decimal places.

Model answer

\(1.5 + 22t - 4.9t^2 > 20\)
\(4.9t^2 - 22t + 18.5 < 0\)
Critical values: \(t = \dfrac{22 \pm \sqrt{484 - 362.6}}{9.8} = \dfrac{22 \pm \sqrt{121.4}}{9.8}\)
\(t = 1.1205...\) or \(t = 3.3692...\)
The ball is above 20 m when \(1.1205... < t < 3.3692...\)
Time \(= 3.3692... - 1.1205... = 2.2486...\)
The ball is more than 20 m above the ground for 2.25 seconds (2 d.p.).

Explanation

The ball is higher than 20 m between the two times when \(h = 20\), so solve the quadratic inequality and find the length of the interval between the critical values. Keep the full calculator values and only round the final difference. A common error is to give one of the times instead of the difference.

Mark scheme

1 mark
P1 for forming the inequality \(1.5 + 22t - 4.9t^2 > 20\)
oe; may use = for this mark; correct answer with no working scores full marks
1 mark
P1 for rearranging to \(4.9t^2 - 22t + 18.5 < 0\)
oe, e.g. \(4.9t^2 - 22t + 18.5 = 0\)
1 mark
M1 for a correct substitution into the quadratic formula, \(\dfrac{22 \pm \sqrt{22^2 - 4 \times 4.9 \times 18.5}}{2 \times 4.9}\)
oe
1 mark
A1 for the critical values 1.12 and 3.37
awrt 1.12 and 3.37 (1.1205... and 3.3692...)
1 mark
A1 for 2.25 seconds
awrt 2.25 (2.2486...)

Ready to try the rest? Every answer you write is marked against the mark scheme, so you can see exactly where the marks are.

AI Tutor Hints first, answers after you try

Hi! Stuck? Ask me for a hint and I’ll help you work it out. Once you’ve checked your answer I can explain exactly where the marks are.