Inequalities

This guide covers subtopic 2.6 Inequalities, from Topic 2, Algebra and graphs, for Cambridge O Level Mathematics (Syllabus D) 4024, 2025–2027 series.

Where this fits in 4024

Inequalities use the same equation-solving moves as Indices and Equations, with one critical extra rule to track (what happens when multiplying or dividing by a negative number), then extends into two-variable graphical regions — useful groundwork for Functions later in the topic.

Syllabus coverage

CAMBRIDGE O LEVEL MATHEMATICS (SYLLABUS D) 4024

• Represent and interpret inequalities, including on a number line (2.6)
• Construct, solve and interpret linear inequalities (2.6)
• Represent and interpret linear inequalities in two variables graphically (2.6)
• List the inequalities that define a given region (2.6)

4024 is not tiered — every candidate covers all of the above. Linear programming problems are not included.

Inequalities on a number line

On a number line, open circles represent strict inequalities (<, >), and closed circles represent inclusive inequalities (⩽, ⩾).

Worked example. Represent −3 ⩽ x < 1 on a number line.

−3 −2 −1  0  1
  ●━━━━━━━━━━○

A closed circle sits at −3 (inclusive, ⩽) and an open circle at 1 (strict, <), with the line shaded solid between them.

Solving linear inequalities

Linear inequalities are solved with the same steps as linear equations, with one rule that has no equation equivalent: multiplying or dividing both sides by a negative number reverses the inequality sign.

Worked example. Solve −3 ⩽ 3x − 2 < 7.

−3 ⩽ 3x − 2 < 7
−1 ⩽ 3x < 9              (add 2 throughout)
−1/3 ⩽ x < 3              (divide by 3 throughout — positive, sign unchanged)

Worked example. Solve 3x < 2x + 4.

3x − 2x < 4
x < 4

Double inequalities and integer solutions

A double inequality — one with three parts, such as −1/3 ⩽ x < 3 above — is solved by operating on all three parts at once.

Worked example. Solve −7 ⩽ 3x + 2 < 11.

−7 ⩽ 3x + 2 < 11
−9 ⩽ 3x      < 9      (subtract 2 from each part)
−3 ⩽ x       < 3      (divide each part by 3)

Because every part of a double inequality was divided by a positive number (3), the direction of both signs stayed the same throughout; had the coefficient of x been negative, both inequality signs would need to be reversed, exactly as in a single inequality.

When a question asks for the integer solutions, apply the endpoints carefully rather than rounding casually: for −3 ⩽ x < 3, the integers are −3, −2, −1, 0, 1, 2 — note that −3 is included (inclusive, ⩽) but 3 is not (strict, <). Miscounting which endpoint is included is a common way to lose an easy mark, particularly under time pressure near the end of a paper.

Inequalities in two variables, graphically

A linear inequality in two variables is represented on a graph as a region bounded by a line: broken lines represent strict inequalities (<, >), solid lines represent inclusive inequalities (⩽, ⩾), and shading marks the unwanted region (unless a question directs otherwise) — leaving the required region visually clear (unshaded).

Worked example. Represent x < 1 and y ⩾ 1 on the same diagram.

Draw a broken vertical line at x = 1 (strict, so broken), shading the unwanted side (x ⩾ 1, to the right); draw a solid horizontal line at y = 1 (inclusive, so solid), shading the unwanted side (y < 1, below it). What’s left unshaded — to the left of x = 1 and on or above y = 1 — is the region satisfying both inequalities.

Listing the inequalities defining a region

Given a shaded diagram, the reverse skill is required: identify each boundary line’s equation, decide whether it should be < / > (broken line) or ⩽ / ⩾ (solid line), and decide which side of each line the unshaded (wanted) region lies on, to state each inequality correctly.

Note: linear programming — optimising a quantity subject to a system of inequalities — is explicitly not included in this syllabus; the skill required stops at representing, solving and reading off regions.

Common mistakes

• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number — the single most common error in this subtopic.
• Mixing up open/closed circles or broken/solid lines. Strict inequalities (<, >) are always the “open” version (open circle, broken line); inclusive inequalities (⩽, ⩾) are always “closed” (closed circle, solid line).
• Shading the wanted region instead of the unwanted one. Unless a question says otherwise, shading marks what’s excluded — the required region is the unshaded part.
• Reading off the wrong side of a boundary line when listing the inequalities that define a shaded region.
• Including or excluding the wrong endpoint when listing integer solutions from a double inequality — check each end separately against its own inequality sign.

Quick revision checklist

• Open vs closed circles on a number line
• Solving linear inequalities, including reversing the sign when multiplying/dividing by a negative
• Solving double inequalities by operating on all three parts at once, and listing integer solutions correctly at each endpoint
• Broken vs solid lines, and which side gets shaded, for two-variable inequalities
• Reading a shaded region back into a list of inequalities
• Linear programming is not included in this syllabus

Related resources

• Indices and Equations — the equation-solving skills this subtopic builds on
• Sequences and Proportion — the next subtopic
• Cambridge O Level Mathematics subject hub

Written against Cambridge O Level Mathematics (Syllabus D) 4024, 2025–2027 series. Always check the current syllabus for your examination year.