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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Inequalities

Linear and quadratic inequalities, set notation, regions on graphs.

12 original questions, 54 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Find the set of values of x for which 5 − 3x < 2x + 20, giving your answer in set notation.

2 marks
Show answer
{x : x > -3}
2

Solve the inequality x2 − 2x − 15 > 0

xy
3 marks
Show answer
x 5
3

Find the set of values of x for which 2x2 + x − 6 ≤ 0, giving your answer in set notation.

3 marks
Show answer
{x : -2 ≤ x ≤ 32}
4

Find the set of values of x for which both 3x − 7 > 2 and x2 − 9x + 14 < 0

4 marks
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3 < x < 7
5

Find the set of values of x for which x2 + 4x − 1 ≥ 0. Give your answer in set notation, using exact values.

3 marks
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{x : x ≤ -2 - √5} ∪ {x : x ≥ -2 + √5}
6

The equation kx2 + 4x + (k − 3) = 0, where k is a non-zero constant, has no real roots.

  1. Show that k2 − 3k − 4 > 0
  2. Hence find the set of possible values of k.
5 marks
Show answer
a) shown b) k 4
7

The diagram shows the curve with equation y = x2 − 2x − 3 and the line with equation y = x + 1. The region R, shown shaded, is bounded by the curve and the line. R does not include its boundary.

  1. Find the coordinates of the points where the line meets the curve.
  2. Use inequalities to define the region R.
xyRy = x + 1y = x² - 2x - 3
5 marks
Show answer
a) (-1, 0) and (4, 5) b) y > x2 - 2x - 3 and y < x + 1
8

The curve C has equation y = 4 − x2 and the line l has equation y = x − 2

  1. Find the coordinates of the points where l meets C.
  2. On the axes, sketch C and l, and shade the region that satisfies both y ≤ 4 − x2 and y ≥ x − 2
xy
5 marks
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a) (-3, -5) and (2, 0) b) region below C and above l, between x = -3 and x = 2
9

The line with equation y = kx − 2, where k is a constant, meets the curve with equation y = x2 + 2x + 2 at two distinct points. Find the set of possible values of k.

5 marks
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k 6
10

Find the set of values of x for which 12x < x + 1, x ≠ 0. Give your answer in set notation.

xy
6 marks
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{x : -4 3}
11

A rectangular patio has length (2x + 3) m and width (x − 1) m. The perimeter of the patio must be more than 30 m and its area must be less than 52 m2.

  1. Show that x > 133
  2. Find the set of possible values of x.
6 marks
Show answer
a) shown b) 133 < x < 5
12

Tomasz volleys a ball in a park in Bristol. The height h metres of the ball above the ground t seconds after it is kicked is modelled by

h = 1 + 12t − 5t2

  1. Find, in set notation, the values of t for which the ball is more than 5 m above the ground.
  2. Find the time at which the model predicts the ball hits the ground. Give your answer to 3 significant figures.
  3. Hence state a suitable range of values of t for which the model is valid.
7 marks
Show answer
a) {t : 0.4 < t < 2} b) 2.48 s c) 0 ≤ t ≤ 2.48

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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