Write down the largest integer value of x that satisfies x < 7.
(Total for Question 9 is 1 mark)
10
Write down all the integer values of x that satisfy -3 < x ≤ 2.
(Total for Question 10 is 2 marks)
11
On the number line below, show the inequality x > -2.
(Total for Question 11 is 1 mark)
12
On the number line below, show the inequality -1 ≤ x < 4.
(Total for Question 12 is 2 marks)
13
The diagram below shows a number line. There is an open circle at x = 3, with shading and an arrow extending to the left. Write down the inequality that is shown.
(Total for Question 13 is 1 mark)
14
Solve 3(x - 2) < 15 Show your working.
(Total for Question 14 is 2 marks)
15
Solve the inequality 2x + 5 < 4x - 3, and show the solution set on the number line below.
(a)Solve the inequality 2x + 5 < 4x - 3.(2)
(b)Show the solution set on the number line below.(1)
(Total for Question 15 is 3 marks)
16
n is an integer such that -4 < 2n ≤ 10. List all the possible values of n.
(Total for Question 16 is 2 marks)
17
Solve the inequality 3(2x - 1) ≥ x + 8, giving your answer using set notation. Show your working.
(Total for Question 17 is 4 marks)
18
A rectangle has length (2x + 3) cm and width x cm. The perimeter of the rectangle is less than 46 cm. Form an inequality in x and solve it to find the range of possible values of x.
(Total for Question 18 is 4 marks)
19
On the grid below, shade the region R that satisfies all three of these inequalities: x ≥ 1, y < 4, y ≥ x. Label your region R.
(Total for Question 19 is 4 marks)
20
Solve the inequality 5 - 3x ≤ 2x - 5, giving your answer using set notation. Show your working.
(Total for Question 20 is 3 marks)
21
Sam has s sweets, where s is a positive integer. Tom has more than 3 times as many sweets as Sam. The total number of sweets that Tom and Sam have together is less than 60. Find the maximum possible value of s.
(Total for Question 21 is 4 marks)
22
Find the set of values of x that satisfy both of these inequalities: 3x - 2 < 10 x + 4 > 6 Show your working.
(Total for Question 22 is 3 marks)
23
Solve the inequality x2 < 49.
(Total for Question 23 is 2 marks)
24
Solve the inequality x2 - 5x - 6 > 0, giving your answer using set notation. Show your working.
(Total for Question 24 is 4 marks)
Mark scheme · 4.5D Inequalities: Fluency and Exam Drill
Question 1
B1 x < 7 oe cao
Answer: x < 7
Question 2
B1 x > 10 oe cao
Answer: x > 10
Question 3
B1 x < 8 oe cao
Answer: x < 8
Question 4
B1 x ≥ 12 oe cao
Answer: x ≥ 12
Question 5
M1 3x ≤ 15 oe (correct rearrangement)
A1 x ≤ 5 cao
Answer: x ≤ 5
Question 6
M1 5x > 25 oe (correct rearrangement)
A1 x > 5 cao
Answer: x > 5
Question 7
M1 4 < 9 + x oe, or -x < 5 oe (correct rearrangement)
A1 x > -5 cao
Answer: x > -5
Question 8
M1 8 ≥ 2x oe (correct rearrangement)
A1 x ≤ 4 cao
Answer: x ≤ 4
Question 9
B1 x = 6 cao
Answer: x = 6
Question 10
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 11
B1 open circle drawn at -2, with shading/arrow extending to the right (all correct)
Answer: Open circle at -2 with shading/arrow to the right
Question 12
B1 closed circle drawn at -1
B1 open circle drawn at 4, with a line segment correctly joining the two circles (no arrows beyond either end)
Answer: Closed circle at -1, open circle at 4, with a line segment joining them
B1 0 < x < 20/3 oe, stating the additional constraint x > 0 since x is a length
Answer: 0 < x < 20/3 (cm)
Question 19
B1 solid vertical line x = 1 drawn correctly, with the region to the right (x ≥ 1) indicated
B1 dashed horizontal line y = 4 drawn correctly, with the region below (y < 4) indicated
B1 solid line y = x drawn correctly, with the region above/left (y ≥ x) indicated
B1 correct triangular region satisfying all three inequalities simultaneously shaded and labelled R
Answer: Triangular region with vertices at (1, 1), (1, 4) [open at this vertex] and (4, 4); solid boundaries on x = 1 and y = x, dashed boundary on y = 4
Question 20
M1 10 ≤ 5x oe (correct rearrangement)
A1 x ≥ 2 cao
B1 {x : x ≥ 2} oe, correct set notation, ft
Answer: {x : x ≥ 2}
Question 21
B1 t > 3s oe, where t is the number of sweets Tom has (correctly represents 'more than 3 times as many')
B1 s + t < 60 oe (correctly represents the total being less than 60)
M1 4s < 60 oe, dependent on both previous marks (correctly combines the two inequalities, e.g. substituting 3s for t)
A1 s = 14 cao, with valid reasoning that s must be a positive integer less than 15
Answer: s = 14
Question 22
M1 x < 4 oe (first inequality solved correctly)
M1 x > 2 oe (second inequality solved correctly)
A1 2 < x < 4 oe cao
Answer: 2 < x < 4
Question 23
M1 critical values x = 7 and x = -7 identified, e.g. from √49 = 7
A1 -7 < x < 7 oe cao
Answer: -7 < x < 7
Question 24
M1 (x - 6)(x + 1) oe (correct factorisation)
A1 critical values x = 6 and x = -1 identified
dM1 correct method to identify the regions satisfying the inequality, dependent on correct critical values, e.g. sign analysis or a sketch of the upward parabola
A1 x < -1 or x > 6 oe cao (accept {x : x < -1} union {x : x > 6}); condone omission of 'or' only if both regions are clearly given