(a)Write down the inequality shown by number line A.(1)
(b)Write down the inequality shown by number line B.(1)
(Total for Question 2 is 2 marks)
3
n is an integer such that -3 < n ≤ 2. List all the possible values of n.
(Total for Question 3 is 2 marks)
4
Solve the inequalities.
(a)x + 6 < 10(1)
(b)5x ≥ 20(1)
(Total for Question 4 is 2 marks)
5
Solve the inequality 2x + 5 < 17.
(Total for Question 5 is 2 marks)
6
Solve the inequality 3x - 4 ≥ 11.
(Total for Question 6 is 2 marks)
7
n is an integer such that -2 ≤ n < 5.
(a)List all the possible values of n.(2)
(b)Write down the smallest possible value of n.(1)
(Total for Question 7 is 3 marks)
8
Solve the inequality 5x + 2 ≤ 3x + 10.
(Total for Question 8 is 3 marks)
9
Solve the inequality 4(x - 1) > 2x + 6.
(Total for Question 9 is 3 marks)
10
(a) Solve the inequality x - 3 > -5. (b) Show the solution set on the number line below.
(a)Solve x - 3 > -5.(1)
(b)Show the solution set on the number line.(2)
(Total for Question 10 is 3 marks)
11
Anjali has p pounds. She has more than 15 pounds but no more than 40 pounds. Write down an inequality, in terms of p, to show this information.
(Total for Question 11 is 2 marks)
12
Tom thinks of a number, x. He multiplies it by 3, then subtracts 7. The result is greater than 14.
(a)Write an inequality, in terms of x, to show this information.(1)
(b)Solve your inequality to find the possible values of x.(2)
(Total for Question 12 is 3 marks)
13
Which of the following is the solution to 2x + 1 ≥ x - 4?
A) x ≥ -5
B) x ≤ -5
C) x ≥ -3
D) x ≤ 5
(Total for Question 13 is 1 mark)
14
Priya is designing a rectangular garden. It has length x metres and width 6 metres. The perimeter of the garden must be at least 30 metres. Form an inequality in x and solve it to find the possible values of x.
Diagram NOT accurately drawn
(Total for Question 14 is 3 marks)
15
Solve the inequality 7 - 2x > 1.
(Total for Question 15 is 2 marks)
16
Solve the inequality 3 - 5x ≤ 18.
(Total for Question 16 is 2 marks)
17
Solve the inequality -3 < 2x + 1 ≤ 9. Show the solution set on the number line below.
(Total for Question 17 is 4 marks)
18
Solve the inequality 3(x - 2) ≥ 5(x + 4), giving your answer in the form x ≤ k, where k is an integer.
(Total for Question 18 is 3 marks)
19
Jamal measures the length of a plank of wood, l cm, as 8 cm, correct to the nearest centimetre. Write down the error interval for l.
(Total for Question 19 is 2 marks)
20
On the grid, shade the region R that satisfies all three of these inequalities: x ≥ 1, y > 2, x + y ≤ 6. Label your region R.
(Total for Question 20 is 3 marks)
21
n is an integer. Show that n = 3 is the only value of n that satisfies both 3n - 1 < 11 and 2n + 3 ≥ 9.
(Total for Question 21 is 4 marks)
Mark scheme · 4.5 Inequalities
Question 1
(a) B1 greater than oe
(a) Answer: greater than
(b) B1 less than or equal to oe
(b) Answer: less than or equal to
Question 2
(a) B1 x > 3 oe cao
(a) Answer: x > 3
(b) B1 x ≤ -2 oe cao
(b) Answer: x ≤ -2
Question 3
M1 -3 correctly excluded and 2 correctly included, at least 3 correct values shown
A1 -2, -1, 0, 1, 2 oe cao, no extra or missing values
Answer: -2, -1, 0, 1, 2
Question 4
(a) B1 x < 4 oe cao
(a) Answer: x < 4
(b) B1 x ≥ 4 oe cao
(b) Answer: x ≥ 4
Question 5
M1 2x < 12 oe, correct rearrangement shown
A1 x < 6 oe cao
Answer: x < 6
Question 6
M1 3x ≥ 15 oe, correct rearrangement shown
A1 x ≥ 5 oe cao
Answer: x ≥ 5
Question 7
(a) M1 -2 correctly included and 5 correctly excluded, at least 5 correct values shown
(a) A1 -2, -1, 0, 1, 2, 3, 4 oe cao, no extra or missing values
(a) Answer: -2, -1, 0, 1, 2, 3, 4
(b) B1 -2 cao, ft from their list in part (a)
(b) Answer: -2
Question 8
M1 terms in x collected on one side, e.g. 2x + 2 ≤ 10 oe
M1 2x ≤ 8 oe, dependent on correct collection of terms
A1 x ≤ 4 oe cao
Answer: x ≤ 4
Question 9
M1 bracket expanded correctly, 4x - 4 > 2x + 6
M1 2x > 10 oe, dependent on correct expansion and collection of terms
A1 x > 5 oe cao
Answer: x > 5
Question 10
(a) B1 x > -2 oe cao
(a) Answer: x > -2
(b) B1 open circle drawn at -2, ft from their answer to part (a)
(b) B1 correct shading/arrow extending to the right from -2, ft from their answer to part (a)
(b) Answer: Open circle at -2 with shading/arrow to the right
Question 11
M1 15 < p oe or p ≤ 40 oe, one boundary correctly represented
A1 15 < p ≤ 40 oe cao, both boundaries correct with correct strict/non-strict signs
Answer: 15 < p ≤ 40
Question 12
(a) B1 3x - 7 > 14 oe cao
(a) Answer: 3x - 7 > 14
(b) M1 3x > 21 oe, ft from their inequality in part (a)
(b) A1 x > 7 oe cao
(b) Answer: x > 7
Question 13
B1 A cao
Answer: A) x ≥ -5
Question 14
M1 2(x + 6) ≥ 30 oe, correct inequality formed for the perimeter
M1 x + 6 ≥ 15 oe, dependent on a correct inequality being formed
A1 x ≥ 9 oe cao
Answer: x ≥ 9
Question 15
M1 -2x > -6 oe, correct rearrangement shown
A1 x < 3 oe cao, inequality sign correctly reversed when dividing by a negative number
Answer: x < 3
Question 16
M1 -5x ≤ 15 oe, correct rearrangement shown
A1 x ≥ -3 oe cao, inequality sign correctly reversed when dividing by a negative number
Answer: x ≥ -3
Question 17
M1 1 subtracted from all three parts, -4 < 2x ≤ 8 oe
dM1 all three parts divided by 2, dependent on the previous method mark, -2 < x ≤ 4 oe
A1 -2 < x ≤ 4 oe cao
B1 correct number line: open circle at -2, closed circle at 4, line joining the two, ft from their algebraic answer
dM1 -2x ≥ 26 oe, dependent on correct expansion, terms in x and constants correctly collected
A1 x ≤ -13 cao, inequality sign correctly reversed when dividing by a negative number
Answer: x ≤ -13
Question 19
B1 7.5 identified as the lower bound
B1 7.5 ≤ l < 8.5 oe cao, correct inequality signs used (lower bound included, upper bound excluded)
Answer: 7.5 ≤ l < 8.5
Question 20
B1 solid vertical line x = 1 drawn correctly with the region to the right (x ≥ 1) indicated
B1 dashed horizontal line y = 2 drawn correctly with the region above (y > 2) indicated
B1 solid line x + y = 6 drawn correctly, and the triangular region satisfying all three inequalities simultaneously shaded and labelled R
Answer: Triangular region with vertices at (1, 2) [open at this corner], (1, 5) and (4, 2); solid boundaries on x = 1 and x + y = 6, dashed boundary on y = 2
Question 21
M1 3n - 1 < 11 solved correctly, n < 4
M1 2n + 3 ≥ 9 solved correctly, n ≥ 3
A1 both inequalities combined correctly, 3 ≤ n < 4 oe
A1 cso, n = 3 stated as the only integer satisfying 3 ≤ n < 4, so n = 3 is the only value (answer printed)