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S3 Maths: Linear Inequalities in One Unknown (Paper 3: challenge)
S3 Maths: Linear Inequalities in One Unknown (Paper 3: challenge)
The third paper puts inequalities back into context. The routine is four steps: define the unknown, form the inequality, solve it, then take the integer answer and write a full sentence.
Direct answers State the unit in the answer, for example 12 kilometres, because a bare number loses marks.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 20 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: S3 Maths: Linear Inequalities in One Unknown |
| Level | S3 Maths |
02 Questions (10 items)
- FoundationSolve the compound inequality 3 < x + 1 <= 7.
- FoundationSolve the compound inequality -1 <= 2x + 3 <= 9.
- FoundationA taxi charges $25 to start and $6 per kilometre. If the fare must not exceed $100, what is the greatest whole number of kilometres?
- IntermediateA rectangle has width x cm and length (x + 4) cm. If its perimeter is less than 36 cm, find the range of x.
- IntermediateThe sum of three consecutive integers is at most 42. What is the greatest possible value of the smallest one?
- IntermediateAt temperature t degrees the change in length of a metal is modelled by 2t – 5 < 15. Find the range of t.
- ChallengeThree times a number plus 7 is greater than 22. Find the smallest integer value of the number.
- ChallengeFive tests must average at least 80 marks. The first four are 76, 82, 79 and 85. What is the minimum mark for the fifth test?
- ChallengeCinema tickets cost $8 each and you have $50. What is the greatest number of tickets you can buy?
- Error redoA student solves 2x + 5 <= 3x – 1 and answers x <= 6. What is wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 2 < x <= 6 | Subtract 1 from all three parts: 3 – 1 = 2 and 7 – 1 = 6, with both signs unchanged. |
| 2 | -2 <= x <= 3 | Subtract 3 everywhere to get -4 <= 2x <= 6, then divide by 2 to get -2 <= x <= 3. |
| 3 | 12 km | Solve 25 + 6k <= 100 to get k <= 12.5, and round down to 12 kilometres. |
| 4 | x < 7 | The perimeter 2(2x + 4) < 36 gives 4x + 8 < 36, so 4x < 28 and x < 7. |
| 5 | 13 | With n as the smallest, 3n + 3 <= 42 gives n <= 13, so the greatest value is 13. |
| 6 | t < 10 | Add 5 to get 2t < 20, then divide by 2 to get t < 10. |
| 7 | 6 | 3x + 7 > 22 gives x > 5, so the smallest integer above 5 is 6. |
| 8 | 78 marks | The first four total 322 and the total needed is 400, so the fifth test needs at least 400 – 322 = 78. |
| 9 | 6 tickets | 8n <= 50 gives n <= 6.25, so round down to 6 tickets at $48. |
| 10 | The correct answer is x >= 6 | Subtracting 2x gives 5 <= x – 1, and adding 1 gives 6 <= x, that is x >= 6. |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
Every challenge question is define the unknown, form the inequality, then take the integer answer. Remember that “at most” rounds down and “at least” rounds up, and state the unit in a full sentence.Tutor Registration →
Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.