Solve the inequality x2 - 4x - 5 < 0. Represent your solution set on the number line below, using open circles to show the boundary values and shading (or hatching) the region that satisfies the inequality.
(Total for Question 6 is 4 marks)
7
n is an integer. n2 - 3n - 10 ≤ 0. Find all the integer values of n that satisfy this inequality.
(Total for Question 7 is 3 marks)
8
Solve the inequality 2x2 - 7x - 4 ≥ 0.
(Total for Question 8 is 4 marks)
9
Solve the inequality 3x2 + x - 2 < 0.
(Total for Question 9 is 4 marks)
10
Solve the inequality x2 - 2x - 15 > 0, giving your answer in set notation.
(Total for Question 10 is 3 marks)
11
The graph of y = x2 - 6x + 8 crosses the x-axis at x = 2 and x = 4, and the curve opens upward (the sketch is. Circle the correct solution set of x2 - 6x + 8 ≤ 0 and give a brief reason for your choice.
A) x ≤ 2 or x ≥ 4
B) 2 ≤ x ≤ 4
C) x < 2 or x > 4
D) 2 < x < 4
(Total for Question 11 is 2 marks)
12
Solve the inequality x(x - 3) ≤ 10.
(Total for Question 12 is 4 marks)
13
Solve the inequality 9 - x2 ≥ 0.
(Total for Question 13 is 3 marks)
14
A rectangular lawn has length (x + 4) metres and width x metres, where x > 0. The area of the lawn must be greater than 45 m2. Form an inequality in x and solve it to find the possible values of x.
Diagram NOT accurately drawn
(Total for Question 14 is 5 marks)
15
Given that x > 0, solve the inequality 4 < x2 < 25.
(Total for Question 15 is 4 marks)
16
Solve the inequality 4x2 - 1 ≤ 0.
(Total for Question 16 is 3 marks)
17
The equation x2 + kx + 9 = 0 has no real roots. Find the range of possible values of k.
(Total for Question 17 is 4 marks)
18
Show that x2 + 6x + 11 > 0 for all real values of x.
(Total for Question 18 is 3 marks)
19
The height, h metres, of a ball above the ground t seconds after it is thrown is modelled by h = 20t - 5t2. Find the range of values of t for which the ball is at least 15 metres above the ground.
(Total for Question 19 is 5 marks)
20
Solve the inequality x2 - 4x - 7 > 0, using the quadratic formula. Give your answer correct to 2 decimal places.
(Total for Question 20 is 5 marks)
21
Find the set of values of x for which x2 - x - 12 ≤ 0 and x2 - 9 > 0.
(Total for Question 21 is 5 marks)
22
Find the set of values of x for which x2 - 5x + 6 > 0 and 2x - 3 < 7.
(Total for Question 22 is 5 marks)
Mark scheme · 8.6 Quadratic Inequalities
Question 1
M1 identifies critical values x = 5 and x = -5
A1 -5 ≤ x ≤ 5 oe
Answer: -5 ≤ x ≤ 5
Question 2
M1 factorises (x-3)(x+3) or finds critical values x = 3, x = -3
A1 critical values x = 3 and x = -3 stated
A1 x < -3 or x > 3 oe
Answer: x < -3 or x > 3
Question 3
M1 factorises (x-4)(x+4) or finds critical values x = 4, x = -4
A1 critical values x = 4 and x = -4 stated
A1 x ≤ -4 or x ≥ 4 oe
Answer: x ≤ -4 or x ≥ 4
Question 4
M1 factorises (x+3)(x-2)
A1 critical values x = -3 and x = 2 stated
A1 -3 ≤ x ≤ 2 oe cao
Answer: -3 ≤ x ≤ 2
Question 5
M1 factorises x(x-5)
A1 critical values x = 0 and x = 5 stated
A1 x ≤ 0 or x ≥ 5 oe
Answer: x ≤ 0 or x ≥ 5
Question 6
M1 factorises (x-5)(x+1)
A1 critical values x = -1 and x = 5 stated
B1 open circles drawn at -1 and 5 on the number line (ft)
B1 correct region between -1 and 5 shaded or hatched (ft)
Answer: -1 < x < 5, with open circles at -1 and 5 and the region between them shaded
Question 7
M1 factorises (n-5)(n+2) or finds critical values n = -2, n = 5
A1 -2 ≤ n ≤ 5 oe
A1 all eight integers listed correctly: -2, -1, 0, 1, 2, 3, 4, 5 (ft from their inequality)
Answer: n = -2, -1, 0, 1, 2, 3, 4, 5
Question 8
M1 attempts to factorise 2x2 - 7x - 4
A1 (2x+1)(x-4) or critical values x = -1/2, x = 4
M1 identifies correct regions outside the roots (coefficient of x2 positive, expression ≥ 0)
A1 x ≤ -1/2 or x ≥ 4 oe
Answer: x ≤ -1/2 or x ≥ 4
Question 9
M1 attempts to factorise 3x2 + x - 2
A1 (3x-2)(x+1) or critical values x = -1, x = 2/3
M1 identifies correct region between the roots (expression < 0)
A1 -1 < x < 2/3 oe
Answer: -1 < x < 2/3
Question 10
M1 factorises (x-5)(x+3) or finds critical values x = -3, x = 5
A1 x < -3 or x > 5 oe
B1 correct set notation {x : x < -3} u {x : x > 5} (ft)
Answer: {x : x < -3} u {x : x > 5}
Question 11
B1 valid reason given, e.g. the curve is below or on the x-axis between the roots
B1 correct option B selected
Answer: B) 2 ≤ x ≤ 4
Question 12
M1 expands to x2 - 3x ≤ 10
M1 rearranges to x2 - 3x - 10 ≤ 0
M1 factorises (x-5)(x+2)
A1 -2 ≤ x ≤ 5 oe
Answer: -2 ≤ x ≤ 5
Question 13
M1 rearranges to x2 ≤ 9
A1 critical values x = -3 and x = 3 stated
A1 -3 ≤ x ≤ 3 oe
Answer: -3 ≤ x ≤ 3
Question 14
M1 forms the inequality x(x+4) > 45 oe
M1 rearranges and attempts to factorise x2 + 4x - 45
A1 (x+9)(x-5) or critical values x = -9, x = 5
A1 x < -9 or x > 5
B1 applies the constraint x > 0 to give the final answer x > 5 (ft)
Answer: x > 5
Question 15
M1 solves x2 > 4 to find critical value x = 2
A1 x > 2, using x > 0 oe
M1 solves x2 < 25 to find critical value x = 5
A1 2 < x < 5 (final answer, ft)
Answer: 2 < x < 5
Question 16
M1 factorises as a difference of two squares (2x-1)(2x+1) or finds critical values x = 1/2, x = -1/2
A1 critical values x = -1/2 and x = 1/2 stated
A1 -1/2 ≤ x ≤ 1/2 oe (accept -0.5 ≤ x ≤ 0.5)
Answer: -1/2 ≤ x ≤ 1/2
Question 17
M1 writes the discriminant k2 - 4(1)(9) oe
M1 sets discriminant < 0, i.e. k2 - 36 < 0
M1 factorises (k-6)(k+6) < 0 or finds critical values k = -6, k = 6
A1 -6 < k < 6 oe
Answer: -6 < k < 6
Question 18
M1 attempts to complete the square on x2 + 6x + 11
A1 (x+3)2 + 2 cao
B1 cso: correct conclusion, e.g. (x+3)2 ≥ 0 for all real x, so (x+3)2 + 2 ≥ 2 > 0 for all real x
Answer: x2 + 6x + 11 = (x+3)2 + 2, and since (x+3)2 ≥ 0 for all real x, the expression is always ≥ 2, so it is always greater than 0.
Question 19
M1 forms the inequality 20t - 5t2 ≥ 15 oe
M1 rearranges to a quadratic ≤ 0 (or ≥ 0) form, e.g. 5t2 - 20t + 15 ≤ 0
M1 simplifies and factorises, e.g. t2 - 4t + 3 ≤ 0 as (t-1)(t-3)
A1 1 ≤ t ≤ 3
B1 ft, correctly interprets in context: the ball is at least 15 m high for 1 ≤ t ≤ 3 seconds
Answer: 1 ≤ t ≤ 3 (seconds)
Question 20
M1 substitutes a=1, b=-4, c=-7 into the quadratic formula correctly
M1 simplifies to x = 2 ± √11 or decimal equivalents
A1 critical values awrt x = 5.32 and x = -1.32
M1 identifies correct regions outside the roots (coefficient of x2 positive, expression > 0)
A1 x < -1.32 or x > 5.32 awrt 2dp
Answer: x < -1.32 or x > 5.32 (2dp)
Question 21
M1 factorises (x-4)(x+3) or finds critical values -3, 4 for the first inequality
A1 -3 ≤ x ≤ 4
M1 factorises (x-3)(x+3) or finds critical values -3, 3 for the second inequality
A1 x < -3 or x > 3
A1 correctly combines both solution sets to give 3 < x ≤ 4
Answer: 3 < x ≤ 4
Question 22
M1 factorises (x-2)(x-3) or finds critical values 2, 3 for the quadratic inequality
A1 x < 2 or x > 3
M1 solves the linear inequality 2x - 3 < 7
A1 x < 5
A1 correctly combines both solution sets to give x < 2 or 3 < x < 5