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Linear Inequalities in One Unknown: Paper 2 Multiple-Choice

Ten original Paper 2 questions on solving linear inequalities, compound inequalities joined by “and” or “or”, least and greatest integers, number lines and the properties of inequalities. Detailed solutions can be requested by email.

10 questions · about 15 minutes · answers below each question

All questions are original, written for Math and AI Academy and modelled on past-paper style. They are not reproductions of HKEAA examination questions.

Question 1 · Modelled on 2014 P2 Q7

The solution of $5 - 2 x < 11$ is

$x < - 3$ .
$x > - 3$ .
$x < 3$ .
$x > 3$ .

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B. $- 2 x < 6$ , and dividing by $- 2$ reverses the sign: $x > - 3$ . Option A forgets the reversal.

Question 2 · Modelled on 2020 P2 Q13

The solution of $4 x + 9 > 1$ and $3 - x \geq - 2$ is

$x > - 2$ .
$x \leq 5$ .
$- 2 < x \leq 5$ .
$x \leq - 2$ or $x > 5$ .

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C. The parts give $x > - 2$ and $x \leq 5$ . “And” keeps the overlap, $- 2 < x \leq 5$ . Options A and B each use only one part.

Question 3 · Modelled on 2017 P2 Q5

The solution of $7 - 3 x > 1$ or $2 x - 3 > 5$ is

$x < 2$ .
$x > 4$ .
$2 < x < 4$ .
$x < 2$ or $x > 4$ .

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D. The parts give $x < 2$ and $x > 4$ . They do not overlap, and “or” keeps both. Option C is the gap between them: the numbers that satisfy neither.

Question 4 · Modelled on 2014 P2 Q7

The solution of $- 2 x < 8 < 3 x - 1$ is

$x > 3$ .
$x > - 4$ .
$- 4 < x < 3$ .
$x < - 4$ .

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A. The chain means $- 2 x < 8$ and $8 < 3 x - 1$ , that is, $x > - 4$ and $x > 3$ . Both hold only when $x > 3$ .

Question 5 · Modelled on 2015 P2 Q6

The solution of $15 + 4 x > 3$ or $9 - 2 x < 5$ is

$x > 2$ .
$- 3 < x < 2$ .
$x > - 3$ .
$x < - 3$ .

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C. The parts give $x > - 3$ and $x > 2$ . “Or” keeps every number in either part, which is $x > - 3$ . Option A would be the answer for “and”.

Question 6 · Modelled on 2019 P2 Q7

The least integer satisfying the compound inequality $- 3 (x - 4) + 2 < 20$ or $\frac{x + 5}{2} > 4$ is

$- 2$ .
$- 1$ .
$3$ .
$4$ .

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B. The parts give $x > - 2$ and $x > 3$ , so the solution is $x > - 2$ . The sign is strict, so $- 2$ itself is out and the least integer is $- 1$ .

Question 7 · Modelled on 2023 P1 Q4

How many positive integers satisfy the compound inequality $\frac{3 x - 1}{4} \leq 5$ and $2 x + 3 > 7$ ?

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A. The solution is $2 < x \leq 7$ , so the integers are $3, 4, 5, 6, 7$ . Counting 2 as well gives 6; counting from 1 gives 7.

Question 8 · Modelled on 2014 P2 Q6

If $a < b$ and $c < 0$ , which of the following must be true?

I. $a + c < b + c$
II. $\frac{a}{c} > \frac{b}{c}$
III. $a c < b c$

I and II only
I and III only
II and III only
I, II and III

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A. Adding $c$ keeps the sign, so I is true. Dividing by the negative $c$ reverses it, so II is true. Multiplying by $c$ also reverses it, giving $a c > b c$ , so III is false.

Question 9 · No direct past-paper model · junior syllabus number line

Which of the following represents the solution of $2 x + 3 \geq 1$ and $5 - x > 3$ ?

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D. The parts give $x \geq - 1$ and $x < 2$ , so the solution is $- 1 \leq x < 2$ : a solid dot at $- 1$ , a hollow dot at $2$ . Option A has the dots the wrong way round.

Question 10 · No direct past-paper model · junior syllabus word problem

Peter has $$200$ . He buys a book for $$56$ and $x$ pens at $$12.5$ each. Find the greatest possible value of $x$ .

$10$
$11$
$12$
$16$

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B. $56 + 12.5 x \leq 200$ gives $x \leq 11.52$ , so he can buy at most $11$ pens. Rounding to 12 would cost $$206$ . Option D forgets the book.

Continue practising this topic: Try the Paper 2 Challenge MC for harder variants, or move to Paper 1 long questions — also available as Paper 1 Challenge.

Where this sits in the syllabus

Linear Inequalities in One Unknown is junior secondary content that the HKDSE keeps examining: a four-mark compound inequality in Paper 1 Section A(1) in most years, and usually one question in Paper 2 Section A. Marks are lost in three places: not reversing the sign when dividing by a negative number, treating “and” as “or”, and miscounting the integers at an end that is or is not included.

Full coverage of this topic, and the rest of the course, is on the Mathematics page. More sets are listed on the practice index.

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