OxfordAQA International GCSE Mathematics: Inequalities on number lines and graphs (9260) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover reference A23 in section 3.2.3 of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1, for exams from May/June 2018 onwards. Questions 1 to 6 are Core content, which can also be assessed on the Extension papers. Questions 7 to 12 are marked (Extension only).

Learn the method first in the study guide or the revision notes. The course hub and the printable checklist show where this unit sits, and the free diagnostics help you choose what to practise next.

Questions

1. Solve 5x + 8 < 3x − 6. Show your solution set on a number line. [3]

2. Solve 3(2x − 5) ≥ 4x + 9. Show your solution set on a number line. [4]

3. Solve (x + 3)/2 > 5 − x. Hence write down the smallest integer that satisfies the inequality. [3]

4. Here is a number line.

   ●━━━━━━━━━━━━━━━━○
−3 −2 −1  0  1  2  3  4  5  6  7

The closed circle is at −1 and the open circle is at 6.

(a) Solve 11 − 3x ≤ 2. [3] (b) Write down the inequality shown on the number line. [2]

5. A taxi company charges a fixed fee of £3.50 plus £1.20 for each whole kilometre travelled. Meera has £20 to spend on one journey.

(a) Write down an inequality for n, the number of kilometres she can travel. [1] (b) Solve your inequality and state the greatest whole number of kilometres she can travel. [3]

6. The integer x satisfies both 2x − 7 < 5 and 3x + 2 ≥ −10. Find the sum of all possible values of x. [4]

7. (Extension only) Solve x² − 2x − 24 < 0. [3]

8. (Extension only) Solve 2x² + 7x − 4 > 0. [4]

9. (Extension only) Solve x² − 6x + 1 ≤ 0. Give the end points of your answer in the form a ± b√2. [4]

10. (Extension only) The region R satisfies all three inequalities

y ≥ 1, y < 2x − 1, x + y ≤ 8

(a) For each of the three boundary lines, state whether it is drawn solid or dashed and give the coordinates of two points on it. [3] (b) Find the number of points with whole-number coordinates that lie inside R. [3]

11. (Extension only) A school buys x tables and y chairs. It buys at least 4 tables. It buys at least twice as many chairs as tables. It buys at most 30 items altogether.

(a) Write down three inequalities in x and y for these conditions. [3] (b) The region satisfying all three inequalities is a triangle. Find the coordinates of its corners. [3] (c) Explain whether buying 5 tables and 9 chairs is allowed. [1]

12. (Extension only) Find all integers x that satisfy both x² − 2x − 15 ≤ 0 and 5 − 2x < 8. [5]

Answers

1. 5x − 3x < −6 − 8, so 2x < −14 [1] x < −7 [1] Open circle at −7 with the line or arrow pointing left [1] Answer: x < −7 [3] Examiner insight: Draw the circle at the boundary value and make the shading go the right way. A circle of the wrong type for a strict inequality misrepresents the answer.

2. 6x − 15 ≥ 4x + 9 [1] 2x ≥ 24 [1] x ≥ 12 [1] Closed circle at 12 with the line or arrow pointing right [1] Answer: x ≥ 12 [4] Examiner insight: Expand the bracket on its own line before collecting terms. Multiplying 3 by −5 and getting +15 is a common slip, and a clear line of working lets you see it.

3. x + 3 > 2(5 − x), so x + 3 > 10 − 2x [1] 3x > 7, so x > 7/3 [1] 7/3 is about 2.33, so the smallest integer is 3 [1] [3] Examiner insight: The integer 2 is the tempting wrong answer. Stop and check that your integer is strictly greater than 7/3, then say which integer you chose.

4. (a) −3x ≤ −9 [1] Divide by −3 and reverse the sign [1] x ≥ 3 [1] (b) The closed circle at −1 means −1 is included, giving −1 ≤ x [1] The open circle at 6 means 6 is excluded, giving x < 6 [1] Answer: −1 ≤ x < 6 [5] Examiner insight: Dividing by a negative is where the sign flips. Writing “reverse the sign” next to the step helps you remember it, and a final answer of x ≤ 3 would be wrong.

5. (a) 3.5 + 1.2n ≤ 20 [1] (b) 1.2n ≤ 16.5 [1] n ≤ 13.75 [1] She can pay only for whole kilometres, so the greatest is 13 km (cost £19.10) [1] [4] Examiner insight: A context question ends with a decision. The inequality answer n ≤ 13.75 is not the stated answer, so round down to a whole number and say what it means.

6. 2x < 12, so x < 6 [1] 3x ≥ −12, so x ≥ −4 [1] The integers are −4, −3, −2, −1, 0, 1, 2, 3, 4, 5 [1] Everything from −4 to 4 cancels in pairs, leaving the sum 5 [1] [4] Examiner insight: List the integers before you add. Starting at −4 (included) and finishing at 5 (not 6) is where a missing or extra value usually creeps in.

7. (x − 6)(x + 4) < 0 [1] Critical values x = 6 and x = −4 [1] The U-shaped curve is below the axis between them: −4 < x < 6 [1] [3] Examiner insight: A quick sketch of the U-shape shows whether you need the middle or the outside. A “less than 0” question gives one range between the roots.

8. (2x − 1)(x + 4) > 0 [2] Critical values x = 1/2 and x = −4 [1] Above the axis outside the roots: x < −4 or x > 1/2 [1] [4] Examiner insight: Write the two parts joined by “or”. A single combined statement such as −4 > x > 1/2 is not valid, because no number can satisfy it.

9. x = (6 ± √(36 − 4)) / 2 = (6 ± √32) / 2 [1] √32 = 4√2, so x = 3 ± 2√2 [1] Below or on the axis between the roots [1] Answer: 3 − 2√2 ≤ x ≤ 3 + 2√2 [1] [4] Examiner insight: The question asks for exact surd form, so a rounded decimal like 0.17 ≤ x ≤ 5.83 would not follow the instruction. Keep the surds all the way through.

10. (a) y = 1 is solid (the inequality has ≥), for example (0, 1) and (5, 1) [1] y = 2x − 1 is dashed (the inequality has <), for example (1, 1) and (3, 5) [1] x + y = 8 is solid (the inequality has ≤), for example (0, 8) and (8, 0) [1] (b) Count by column, using y ≥ 1, y < 2x − 1 and y ≤ 8 − x [1]

x	Whole-number y values	Count
1	none (y < 1 and y ≥ 1 clash)	0
2	1, 2	2
3	1, 2, 3, 4	4
4	1, 2, 3, 4	4
5	1, 2, 3	3
6	1, 2	2
7	1	1

All columns correct [1] Total: 2 + 4 + 4 + 3 + 2 + 1 = 16 points [1] [6] Examiner insight: Points on a dashed line are not in the region, so (3, 5) is out, and points on a solid line are in, so (4, 4) on x + y = 8 counts. A table keeps the working easy to check.

11. (a) x ≥ 4 [1] y ≥ 2x [1] x + y ≤ 30 [1] (b) x = 4 meets y = 2x at (4, 8) [1] x = 4 meets x + y = 30 at (4, 26) [1] y = 2x meets x + y = 30: 3x = 30, so x = 10, y = 20, giving (10, 20) [1] (c) With 5 tables, y must be at least 10. Nine chairs is fewer than 10, so it breaks “y ≥ 2x” and is not allowed [1] [7] Examiner insight: Turn each phrase into a symbol first: “at least” is ≥ and “at most” is ≤. Then answer part (c) by substituting into the inequality that fails and say which one.

12. (x − 5)(x + 3) ≤ 0 [1] −3 ≤ x ≤ 5 [1] −2x < 3 [1] Divide by −2 and reverse the sign: x > −3/2 [1] Both conditions: −3/2 < x ≤ 5, so the integers are −1, 0, 1, 2, 3, 4, 5 [1] [5] Examiner insight: When two conditions apply, draw both on number lines and shade only the overlap. Include 5, because the closed end of the first condition keeps it, and exclude −2, because it lies outside the second.

Where marks are usually lost

• Leaving the inequality sign unchanged after a division or multiplication by a negative number.
• Mixing up open and closed circles, so a strict inequality gets a closed circle.
• Answering with an equation such as x = 12 instead of an inequality.
• Counting an integer that sits on an open end, or leaving out one on a closed end.
• Forgetting to round down to a whole number in a context question about items that can only be whole.
• Giving a single range for “outside the roots” quadratic answers, or the wrong joining word between the two parts.
• Rounding surds early when an exact answer is requested.
• Drawing a solid line for a strict inequality or a dashed line for an included one.
• Testing a point that is on the boundary line, so the region test tells you nothing.

Next steps

• Review the method in the revision notes or the study guide.
• Visit the OxfordAQA International GCSE Mathematics hub and tick off topics on the printable checklist.
• Try all the free 10-minute diagnostics.
• Book a free trial class if you want help from a teacher.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), Version 5.1 (for exams May/June 2018 onwards), published by OxfordAQA. Section 3.2.3, reference A23.