GCSE Maths · Higher · Chapter 15: Equations and inequalities · Lesson 3

Linear and graphical inequalities

Grades 4–8About 50 minutesCalculator allowed40 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/gcse-linear-and-graphical-inequalities/

NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Solve $4 (x - 1) = 16$ .
[1 mark]
Answer
A2
Last lesson
Solve $x + y = 12$ and $x - y = - 2$ .
[1 mark]
Answer
A3
From chapter 5
A force of 35 N acts on 5 m $^{2}$ . Find the pressure.
[1 mark]
Answer
A4
From chapter 10
A line has gradient $\frac{1}{4}$ . Write the gradient of a line perpendicular to it.
[1 mark]
Answer

Grades 4–6

Your teacher works through each example with you. Then try the one beside it.

Example 1

Solve $4 x + 9 \leq 2 x + 21$ .

Hint Take

2 x

from both sides, as in an equation.

Example 2

Solve $5 - 4 x > 17$ .

Hint Dividing by a negative number turns the sign round.

Example 3

Solve $- 13 < 3 x - 4 \leq 8$ . List the integers that satisfy it.

Hint Do the same to all three parts.

B3
$x$ is an integer and $- 5 \leq \frac{x + 1}{2} < 3$ . How many possible values of $x$ are there?
[3 marks]
Answer

Grades 5–7

C1
The number line shows the solution of $3 x - 2 > k$ . Work out $k$ .
[2 marks]
Answer
C2
Solve $\frac{2 x + 5}{3} < x - 1$ .
[3 marks]
Answer

C3
Find the largest integer $n$ such that $4 n + 3 > 7 n - 20$ .
[2 marks]
Answer
C4
Ali thinks of a whole number. Three times it, take away $5$ , is more than $20$ . Twice it is less than $22$ . What could it be?
[3 marks]
Answer
C5
$R$ is the region where $y > 2 x + 1$ and $x + y < 8$ . Is the point $(2, 5)$ in $R$ ? Give a reason.
[2 marks]

Grades 6–8

D1
Priya says, "No integer satisfies $- 3 \leq 4 - x < 2$ . When I take $4$ off every part I get $- 7 \leq - x < - 2$ , so $7 \leq x < 2$ , and no number can be both."
Explain her mistake. Then list all the integers that do satisfy the inequality.
[3 marks]

D2
The length of this rectangle is greater than its width, and its perimeter is at most $50$ cm. $x$ is a whole number. Find all the possible values of $x$ .
[4 marks]
Not drawn accurately
Answer
D3
The region $R$ is shaded on the grid. One of its three boundary lines is dashed.
(a)
Write down the three inequalities that define $R$ .
[3 marks]
Answer
(b)
The point $(k, 4)$ is in $R$ . Find the range of possible values of $k$ .
[2 marks]
Answer
Total for D3: 5 marks

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve an inequality, turning the sign round for a negative.
I can list the integers a double inequality allows.
I can describe a region with inequalities.

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.