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Mathematics questions, chapter 6
Nama / NameKelas / ClassTarikh / Date

Linear Inequalities in Two Variables

45 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Show your working for Section B.

Section A: objective questions

12 marks
Choose the correct answer. Each question carries 1 mark.
1

Question 1

How is the boundary line drawn, and which side is shaded, for the inequality y ≥ mx + c?
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B
≥ includes the points on the line, so the line is solid. y ≥ means on and above the line. A dashed line is for > and < only.
For the teacher: Remembering, low difficulty, standard 6.1.3
Textbook p. 162
2

Question 2

A pen costs RM3 and a notebook costs RM5. Dina can spend at most RM60 on x pens and y notebooks. Which inequality represents this?
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C
Total cost is 3x + 5y and it must not be more than 60, so ≤. The last option swaps the prices.
For the teacher: Understanding, low difficulty, standard 6.1.1
Textbook p. 156
3

Question 3

Which point satisfies y < 3x − 2?
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A
At (0, −3): 3(0) − 2 = −2 and −3 < −2, true. At (1, 2): 3(1) − 2 = 1 and 2 > 1. (2, 4) lies on the line (4 = 4). At (−1, −4): 3(−1) − 2 = −5 and −4 > −5.
For the teacher: Applying, medium difficulty, standard 6.1.2
Textbook p. 160
4

Question 4

Diagram 1 shows a dashed line passing through (0, 2) and (4, 0). The region below it is shaded. Which inequality is represented?
Dashed line through (0, 2) and (4, 0) with the region below the line shaded
Diagram 1
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D
Gradient = (0 − 2) ÷ (4 − 0) = −½ and the y-intercept is 2, so the line is y = −½x + 2. Dashed means strict, and below means less than: y < −½x + 2.
For the teacher: Applying, medium difficulty, standard 6.1.3
Textbook p. 162
5

Question 5

The inequality x − y > 4 is the same as
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C
Subtract x from both sides: −y > 4 − x. Multiply by −1 and reverse the sign: y < x − 4. The first option forgets to reverse the sign.
For the teacher: Applying, medium difficulty, standard 6.1.3
Textbook p. 163
6

Question 6

The number of boys y is not more than three times the number of girls x. Which inequality shows this?
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A
'Not more than' means ≤ and 'three times the number of girls' is 3x. x ≤ 3y swaps the two quantities, and < leaves out the equal case.
For the teacher: Remembering, low difficulty, standard 6.2.1
Textbook p. 165
7

Question 7

y exceeds x by at least 6. Which inequality represents this?
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D
The difference y − x is 6 or more, so y − x ≥ 6. 'At least' includes 6, so > is too strict. y ≥ 6x means 6 times, not 6 more.
For the teacher: Understanding, medium difficulty, standard 6.2.1
Textbook p. 165
8

Question 8

Which point satisfies both x + y ≤ 8 and y ≥ 3?
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B
(3, 4): 3 + 4 = 7 ≤ 8 and 4 ≥ 3, both true. (5, 2) and (1, 2) fail y ≥ 3. (6, 4) has 10 > 8.
For the teacher: Applying, low difficulty, standard 6.2.2
Textbook p. 167
9

Question 9

Diagram 2 shows a shaded region. The side joining (6, 0) and (0, 6) is a dashed line. Which three inequalities define the region?
Triangle with corners (0, 0), (6, 0) and (0, 6); the slanting side is dashed and the sides on the axes are solid
Diagram 2
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D
The slanting line has gradient −1 and passes through (0, 6): x + y = 6. It is dashed and the region is below it, so x + y < 6. The axes are solid and the region is above the x-axis and to the right of the y-axis: y ≥ 0 and x ≥ 0.
For the teacher: Understanding, medium difficulty, standard 6.2.4
Textbook p. 171
10

Question 10

Which point does not satisfy the system y > x, y ≤ 5 and x ≥ 0?
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A
(3, 3) has y = x, but the system needs y > x, so it fails. The points (1, 4), (0, 5) and (2, 5) satisfy all three inequalities.
For the teacher: Analysing, medium difficulty, standard 6.2.2
Textbook p. 167
11

Question 11

A choir has x girls and y boys. There are at most 20 pupils, and the number of girls is at least twice the number of boys. Which is a possible choir?
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C
The conditions are x + y ≤ 20 and x ≥ 2y. For x = 12, y = 6: 18 ≤ 20 and 12 ≥ 12, both true. (10, 8) has 10 < 16. (14, 8) has 22 > 20. (6, 10) has 6 < 20.
For the teacher: Applying, medium difficulty, standard 6.2.4
Textbook p. 173
12

Question 12

Diagram 3 shows the triangle that satisfies y ≥ x, y ≤ 6 and x ≥ 0. Which point is in the shaded region?
Triangle with corners (0, 0), (0, 6) and (6, 6) shaded, all sides solid
Diagram 3
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B
(3, 5): 5 ≥ 3, 5 ≤ 6 and 3 ≥ 0, all true. (2, 1) has y < x. (7, 6) has y < x. (−1, 2) has x < 0.
For the teacher: Applying, medium difficulty, standard 6.2.3
Textbook p. 169

Section B: structured questions

33 marks
Answer all questions. Show your working.
13

Question 13

Write a linear inequality for each situation. Let x and y stand for the two items.
[1 mark] (a)Rice costs RM8 per kg (x) and sugar costs RM6 per kg (y). Madam Salmah spends at most RM100.
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Tick each point you got.
[2 marks] (b)A cake costs RM2.50 (x) and a cookie costs RM1.50 (y). A stall wants takings of more than RM90. Write it without decimals.
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Multiplying both sides by 2 removes the decimals and the sign stays the same.
Tick each point you got.
For the teacher: Applying, low difficulty, standard 6.1.1
Textbook p. 157
14

Question 14

Diagram 4 shows a shaded region bounded by a solid line that passes through (0, 4) and (2, 0).
Solid line through (0, 4) and (2, 0) with the region below the line shaded
Diagram 4
[2 marks] (a)Find the equation of the line.
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Tick each point you got.
[2 marks] (b)State the inequality that defines the shaded region.
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For the teacher: Applying, medium difficulty, standard 6.1.3
Textbook p. 162
15

Question 15

A school hall committee has x girls and y boys. The committee has at most 30 members. There are at least 5 boys. The number of girls is not less than the number of boys.
[1 mark] (a)Write the inequality for the size of the committee.
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Tick each point you got.
[1 mark] (b)Write the inequality for the number of boys.
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[1 mark] (c)Write the inequality comparing girls and boys.
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For the teacher: Creating, medium difficulty, standard 6.2.1
Textbook p. 166
16

Question 16

Diagram 5 shows the lines x + y = 6 and y = 2, which divide the plane into regions A, B, C and D. The point Q is (3, 4).
Lines x + y = 6 and y = 2 on a grid. Region A is above y = 2 and below x + y = 6, B is above both lines, C is below y = 2 and above x + y = 6, D is below both lines. Point Q is in region B.
Diagram 5
[1 mark] (a)Which region satisfies x + y ≤ 6 and y ≥ 2?
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Tick each point you got.
[1 mark] (b)Which region satisfies x + y ≥ 6 and y ≤ 2?
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[2 marks] (c)Does Q satisfy x + y ≤ 6 and y ≥ 2? Show your working.
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Tick each point you got.
For the teacher: Analysing, medium difficulty, standard 6.2.2
Textbook p. 167
17

Question 17

Diagram 6 shows a triangle with corners (0, 0), (6, 0) and (3, 3). The side from (0, 0) to (3, 3) is a dashed line. The other two sides are solid.
Triangle with corners (0, 0), (6, 0) and (3, 3) shaded; the side from the origin to (3, 3) is dashed
Diagram 6
[2 marks] (a)Find the equation of the line through (6, 0) and (3, 3).
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Tick each point you got.
[3 marks] (b)Write three inequalities that define the shaded region.
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The dashed side y = x gives a strict inequality. The shaded triangle is below that line and above the x-axis.
Tick each point you got.
For the teacher: Applying, medium difficulty, standard 6.2.4
Textbook p. 172
18

Question 18

A school trip has x pupils and y teachers. There are at most 24 people. The number of pupils is at least 3 times the number of teachers. Diagram 7 shows the common region for the system.
Triangle with corners (0, 0), (24, 0) and (18, 6) shaded on axes where x is pupils and y is teachers
Diagram 7
[2 marks] (a)Write two inequalities, other than x ≥ 0 and y ≥ 0, for the conditions.
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Tick each point you got.
[1 mark] (b)From the graph, state the maximum number of teachers.
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[2 marks] (c)State the least and greatest number of pupils when there are 4 teachers.
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[1 mark] (d)Can there be 8 teachers? Give a reason.
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With y = 8 the pupils would need x ≥ 24 and x ≤ 16 together, which is impossible.
Tick each point you got.
For the teacher: Analysing, high difficulty, standard 6.2.4
Textbook p. 173
19

Question 19

A florist makes x bouquets of type A and y bouquets of type B each day. The conditions are: at most 30 bouquets in total, at most 20 bouquets of type A, and at least 5 bouquets of type B. The profit is RM10 for a bouquet of type A and RM8 for a bouquet of type B.
[3 marks] (a)Write three inequalities, other than x ≥ 0 and y ≥ 0, for the conditions.
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Tick each point you got.
[2 marks] (b)If 15 bouquets of type A are made, find the least and greatest number of type B.
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[2 marks] (c)Find the greatest profit when 15 bouquets of type A are made.
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[1 mark] (d)A helper says that making 12 of type A and 20 of type B follows the conditions. Is he right? Justify your answer.
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Tick each point you got.
For the teacher: Evaluating, high difficulty, standard 6.2.4
Textbook p. 174