IGCSE · Practice questions · Linear inequalities and the number line

Find the range.

Six original Cambridge-style questions on inequalities: reading and drawing number lines, listing integers, solving (including reversing the sign), and writing inequalities from a context.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Treat it like an equation, watch the sign.

Number line: open circle for \(<\) or \(>\); closed circle for \(\le\) or \(\ge\).
Solve: do the same to every side, exactly as for an equation.
Negative: multiplying or dividing by a negative number reverses the sign.
Double inequality: work on all three parts at once.
Integers: check each end point: is it included or not?

Recall

[2 marks]

(a) Describe how to show \(-1 \le x < 3\) on a number line.

(b) A number line shows an open circle at 5 with an arrow pointing left. Write down the inequality.

(a) Closed circle at \(-1\), open circle at 3, joined by a line. ✓
(b) \(x < 5\) ✓

Recall

[1 marks]

List the integers \(n\) such that \(-4 \le n < 1\).

\(-4, -3, -2, -1, 0\) ✓

Calculation

[2 marks]

(Extended) Solve \(4x - 9 > x + 6\).

Subtract \(x\): \(3x - 9 > 6\). ✓
Add 9: \(3x > 15\), so \(x > 5\). ✓

Calculation

[2 marks]

(Extended) Solve \(2 - 5x \ge 17\).

Subtract 2: \(-5x \ge 15\). ✓
Divide by \(-5\) and reverse the sign: \(x \le -3\). ✓

Analysis

[3 marks]

(Extended) Solve \(-7 < 2x - 3 \le 5\) and list the integer values of \(x\).

Add 3 to all parts: \(-4 < 2x \le 8\). ✓
Divide by 2: \(-2 < x \le 4\). ✓
Integers: \(-1, 0, 1, 2, 3, 4\). ✓

Analysis

[4 marks]

(Extended) A rectangle is \((x + 6)\) cm long and 4 cm wide.

Its perimeter must be at least 30 cm and its area must be less than 52 cm².

(a) Write down two inequalities in \(x\) and solve them. (b) \(x\) is an integer. Find the possible values of \(x\).

Perimeter: \(2(x + 6) + 2 \times 4 = 2x + 20 \ge 30\), so \(2x \ge 10\) and \(x \ge 5\). ✓
Area: \(4(x + 6) < 52\), so \(x + 6 < 13\) and \(x < 7\). ✓
Together: \(5 \le x < 7\). ✓
(b) \(x = 5\) or \(x = 6\). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.