On the number line below, show the inequality -3 ≤ x < 2.
(Total for Question 2 is 2 marks)
3
A region R on a graph is defined by the inequality y < 2x + 1. For each point below, use substitution to state whether the point lies inside region R.
(a)A(1, 2)(1)
(b)B(3, 8)(1)
(c)C(-1, -3)(1)
(Total for Question 3 is 3 marks)
4
Solve the inequality 3x - 7 ≤ 2x + 5, and show the solution set on the number line below.
(Total for Question 4 is 3 marks)
5
(a) Draw the graph of x + y = 4 on the grid. (b) By shading the unwanted region, show the region that satisfies x + y ≤ 4.
(a)Draw the graph of x + y = 4.(2)
(b)Shade the region that satisfies x + y ≤ 4, labelling it R.(2)
(Total for Question 5 is 4 marks)
6
The diagram shows a straight line and a shaded region. Write down the inequality that defines the shaded region.
(Total for Question 6 is 2 marks)
7
The diagram shows a solid straight line through (0, 3) and (3, 0), with the region above the line shaded. Which inequality is represented by the shaded region?
A) y ≤ -x + 3
B) y ≥ -x + 3
C) y ≤ x - 3
D) y ≥ x - 3
(Total for Question 7 is 1 mark)
8
(a) On the grid, draw the graphs of x = 1, y = 2 and x + y = 6. (b) Shade the region R that satisfies all three inequalities x ≥ 1, y ≥ 2 and x + y ≤ 6.
(a)Draw the graphs of x = 1, y = 2 and x + y = 6.(3)
(b)Shade the region R that satisfies x ≥ 1, y ≥ 2 and x + y ≤ 6.(2)
(Total for Question 8 is 5 marks)
9
Using the region R from Question 8 (defined by x ≥ 1, y ≥ 2 and x + y ≤ 6), write down the coordinates of all the points with integer coordinates that lie inside or on the boundary of R.
(Total for Question 9 is 3 marks)
10
The diagram shows a shaded triangular region R. Write down the three inequalities that define region R.
(Total for Question 10 is 3 marks)
11
The grid shows the region R defined by 2 ≤ x ≤ 5 and 2 ≤ y ≤ x, shaded. Calculate the area of region R.
(Total for Question 11 is 3 marks)
12
Solve the inequality x2 > 16.
(Total for Question 12 is 2 marks)
13
Using the region R from Question 10 (defined by x ≥ 2, y < 5 and y ≥ x + 1), list all the points with integer coordinates that lie inside R, and hence find the greatest possible value of x + y for a point in R.
(Total for Question 13 is 4 marks)
14
By factorising, solve the inequality x2 - x - 6 ≤ 0.
(Total for Question 14 is 3 marks)
15
The curve y = 5 - x2 and the line y = 1 are drawn on the grid. (a) Shade the region R that satisfies both y ≤ 5 - x2 and y ≥ 1. (b) Find the coordinates of the two points where the boundaries of R meet.
(a)Shade the region R that satisfies both y ≤ 5 - x2 and y ≥ 1.(3)
(b)Find the coordinates of the two points where the boundaries of R meet.(2)
(Total for Question 15 is 5 marks)
16
The circle x2 + y2 = 9 is drawn on the grid, together with the positive x-axis and positive y-axis. Shade the region that satisfies x2 + y2 ≤ 9, x ≥ 0 and y ≥ 0.
(Total for Question 16 is 4 marks)
17
The curve y = 4 - x2, the line x = -1 and the line y = 0 are drawn on the grid. Shade the region R that satisfies all three inequalities y ≤ 4 - x2, x ≥ -1 and y ≥ 0.
(Total for Question 17 is 4 marks)
18
A region R satisfies both y ≤ 6 - x2 and y ≥ x + 4. Show that a point with x-coordinate x can only lie in R when -2 ≤ x ≤ 1.
(Total for Question 18 is 4 marks)
Mark scheme · 6.7 Inequalities on Graphs
Question 1
B1 -2 < x seen or implied (open circle correctly interpreted as strict inequality) oe
B1 x ≤ 4 seen or implied (closed circle correctly interpreted as non-strict inequality), combined correctly as -2 < x ≤ 4 cao
Answer: -2 < x ≤ 4
Question 2
B1 closed (filled) circle at -3 oe
B1 open (unfilled) circle at 2 with a line joining -3 to 2 oe
Answer: Filled circle at -3, unfilled circle at 2, joined by a bold line.
Question 3
(a) B1 2(1) + 1 = 3 and 2 < 3, so A lies inside R oe
(a) Answer: Yes, A lies inside R
(b) B1 2(3) + 1 = 7 and 8 is not less than 7, so B does not lie inside R oe
(b) Answer: No, B does not lie inside R
(c) B1 2(-1) + 1 = -1 and -3 < -1, so C lies inside R oe
(c) Answer: Yes, C lies inside R
Question 4
M1 collects x terms and number terms on correct sides, e.g. 3x - 2x ≤ 5 + 7
A1 x ≤ 12 oe cao
B1 closed circle at 12 with line/arrow extending to the left (ft their critical value)
Answer: x ≤ 12
Question 5
(a) M1 correct points found and plotted, e.g. (0, 4) and (4, 0)
(a) A1 correct straight line drawn through (0, 4) and (4, 0), solid (since inequality is non-strict)
(a) Answer: Straight solid line through (0, 4) and (4, 0)
(b) M1 tests a point not on the line, e.g. (0,0): 0 + 0 = 0 ≤ 4 is true, so the origin's side is required
(b) A1 region containing the origin (below/left of the line) shaded and labelled R, solid boundary
(b) Answer: Region below/left of the line x + y = 4, containing the origin, shaded and labelled R
Question 6
B1 recognises the dashed line means a strict inequality (< or >)
B1 y < 2x - 1 cao (correct equation of the line with correct direction of shading)
Answer: y < 2x - 1
Question 7
B1 B cao
Answer: B
Question 8
(a) B1 correct vertical line x = 1 drawn, solid
(a) B1 correct horizontal line y = 2 drawn, solid
(a) B1 correct line x + y = 6 drawn through (0, 6) and (6, 0), solid
(a) Answer: Three solid lines: x = 1, y = 2, and x + y = 6 through (0,6) and (6,0)
(b) M1 tests a point, e.g. (2, 3): 2 ≥ 1 true, 3 ≥ 2 true, 2 + 3 = 5 ≤ 6 true, so (2,3) is inside R
(b) A1 correct triangular region shaded (bounded by the three lines) and labelled R, all boundaries solid
(b) Answer: Triangular region with vertices (1,2), (1,5), (4,2), shaded and labelled R
Question 9
B1 correct points for x = 1: (1,2), (1,3), (1,4), (1,5)
B1 correct points for x = 2 and x = 3: (2,2), (2,3), (2,4), (3,2), (3,3)
B1 correct point for x = 4: (4,2), with no extra or incorrect points listed cao
B1 y ≥ x + 1 (solid sloped edge, non-strict, correct equation)
Answer: x ≥ 2, y < 5, y ≥ x + 1
Question 11
M1 identifies the shape as a right-angled triangle with vertices (2,2), (5,2) and (5,5)
M1 correct method, area = 0.5 x base x height = 0.5 x 3 x 3
A1 4.5 (units squared) cao
Answer: 4.5 square units
Question 12
M1 identifies critical values x = 4 and x = -4
A1 x < -4 or x > 4 oe cao
Answer: x < -4 or x > 4
Question 13
B1 identifies that y < 5 restricts integer y to y ≤ 4
B2 correct list of all three integer points (2,3), (2,4), (3,4), with no incorrect points included (B1 for at least one correct point with no incorrect points)
B1 greatest value of x + y = 7, achieved at (3,4) cao
Answer: Points: (2,3), (2,4), (3,4); greatest value of x + y is 7
Question 14
M1 correct factorisation (x - 3)(x + 2)
A1 critical values x = 3 and x = -2 identified
A1 -2 ≤ x ≤ 3 oe cao
Answer: -2 ≤ x ≤ 3
Question 15
(a) M1 identifies the region below/on the curve (y ≤ 5 - x2)
(a) M1 identifies the region above/on the line (y ≥ 1)
(a) A1 correct lens-shaped region shaded and labelled R, both boundaries solid
(a) Answer: Lens-shaped region between y=1 and the curve y=5-x2, shaded and labelled R, solid boundaries
(b) M1 sets 5 - x2 = 1 and rearranges to x2 = 4
(b) A1 (-2, 1) and (2, 1) cao
(b) Answer: (-2, 1) and (2, 1)
Question 16
M1 identifies the region inside/on the circle (x2 + y2 ≤ 9)
B1 identifies x ≥ 0 (right of the y-axis)
B1 identifies y ≥ 0 (above the x-axis)
A1 correct quarter-disc region in the first quadrant shaded, all boundaries solid
Answer: Quarter-disc region in the first quadrant, bounded by the circle of radius 3 centred at the origin, the positive x-axis and the positive y-axis, all boundaries solid
Question 17
M1 identifies the region below/on the curve (y ≤ 4 - x2)
M1 identifies the region right of/on the line x = -1
M1 identifies the region above/on the line y = 0
A1 correct region shaded and labelled R (bounded by x = -1 on the left, the x-axis below, and the curve above), all boundaries solid
Answer: Region bounded on the left by x=-1, below by y=0, and above by y=4-x2, shaded and labelled R, all boundaries solid
Question 18
M1 combines the inequalities correctly, e.g. x + 4 ≤ 6 - x2