Robinson Tuition
Year 9 · Key Stage 3 · free KS3 maths worksheet

Linear inequalities

12 original questions, 33 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1
  1. Write each statement as an inequality.
    (i)  x is greater than 5
    (ii)  n is at most 10
    (iii)  y is at least −2
  2. Explain the difference between x < 4 and x ≤ 4
3 marks
Show answer
a) x > 5, n ≤ 10, y ≥ -2 b) x < 4 means less than 4, so 4 is not included; x ≤ 4 means less than or equal to 4, so 4 is included
2

n is an integer.

−2 < n ≤ 3

Write down all the possible values of n.

2 marks
Show answer
-1, 0, 1, 2, 3
3

The number line shows an inequality.

  1. Write down the inequality shown, using x.
  2. Write down the smallest integer that satisfies the inequality.
−5−4−3−2−101234
2 marks
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a) x > -2 b) -1
4

On the number line, show the inequality  −1 < x ≤ 3

−4−3−2−1012345
2 marks
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Open circle at -1, solid circle at 3, joined by a line
5

The diagram shows an inequality on a number line.

Write down the inequality shown, using x.

−3−2−10123456
2 marks
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-1 <= x < 4
6

Solve 4x + 3 < 19

2 marks
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x < 4
7

A taxi charges £4 plus £2 for every mile.

Leo has £15. He travels m miles.

  1. Write an inequality in m to show that Leo cannot spend more than £15.
  2. Solve your inequality to find the greatest whole number of miles Leo can travel.
3 marks
Show answer
a) 4 + 2m ≤ 15 b) 5
8
  1. Solve 3(x − 1) ≥ 6
  2. Show your solution on the number line.
−2−1012345678
3 marks
Show answer
a) x >= 3 b) Solid circle at 3 with a line and arrow pointing right
9

Solve 5x − 2 > 3x + 10

3 marks
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x > 6
10
  1. Solve −3 < 2x + 1 ≤ 9
  2. Write down all the integer values of x that satisfy your answer to part a.
4 marks
Show answer
a) -2 < x <= 4 b) -1, 0, 1, 2, 3, 4
11
  1. Solve  5 ≤ 3x − 4 < 14
  2. Show your answer to part a on the number line.
012345678
3 marks
Show answer
a) 3 ≤ x < 6 b) filled circle at 3, open circle at 6, joined by a line
12

Ayesha thinks of a whole number, n.

She doubles it and adds 3.
Her answer is greater than 10 and less than or equal to 21.

  1. Write a two-sided inequality in n.
  2. Work out all the possible values of n.
4 marks
Show answer
a) 10 < 2n + 3 ≤ 21 b) 4, 5, 6, 7, 8, 9

Your score

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Every question here is original and written by Robinson Tuition. It follows the Key Stage 3 programme of study; Year 9 sheets end with extension questions for top sets.

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