Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want Linear Equations and Inequalities — solving for unknowns and representing inequality solutions — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise Linear Equations and Inequalities in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the Linear Equations and Inequalities revision-guide angle, while Tutopiya’s Linear Equations and Inequalities subtopic page owns the learning resource and the free Linear Equations and Inequalities quiz owns the practice.

Linear Equations and Inequalities are central to Algebra in Cambridge IGCSE Mathematics (0580/0607). Examiners test solving one-step and multi-step equations, equations with brackets and fractions, and linear inequalities — often asking you to show the solution on a number line. The methods mirror changing the subject: undo operations in reverse order. This guide explains the subtopic, exam wording, and where to practise.

Key takeaways

• An equation has an equals sign — solve to find the exact value(s) of the unknown.
• An inequality uses <, >, ≤ or ≥ — the solution is a range of values.
• When multiplying or dividing an inequality by a negative number, reverse the inequality sign.
• This subtopic builds on Simplifying Algebraic Expressions and Factorisation.

What are Linear Equations and Inequalities in Cambridge IGCSE Maths?

A linear equation is an equation where the highest power of the unknown is 1 — e.g. 3x − 7 = 11. A linear inequality replaces = with <, >, ≤ or ≥. In Cambridge IGCSE Mathematics, you solve by isolating the unknown, simplify expressions on each side first, and for inequalities represent the answer on a number line with open or closed circles. Word problems translate into equations before solving.

Read the full explanation on Tutopiya’s Linear Equations and Inequalities subtopic page before you attempt questions.

Equations vs inequalities: what changes

Feature	Equation	Inequality
Symbol	=	<, >, ≤, ≥
Solution	Usually one value	A range of values
Multiply/divide by negative	No sign change	Reverse inequality sign
Number line	Single point	Ray or segment; ○ open, ● closed

How to solve linear equations — step by step

1. Expand brackets and collect like terms on each side.
2. Move unknown terms to one side (usually left) and numbers to the other.
3. Add/subtract to isolate the term containing the unknown.
4. Divide by the coefficient of the unknown.
5. Check by substituting back into the original equation.

Test yourself with the free Linear Equations and Inequalities quiz.

How to solve linear inequalities — step by step

1. Solve like an equation — expand, collect, isolate.
2. If you multiply or divide by a negative, flip the sign: −2x < 6 → x > −3.
3. Write the solution — e.g. x ≥ 4 or −1 < x ≤ 5.
4. Draw on a number line: open circle for < or >; closed circle for ≤ or ≥.
5. Shade the correct direction — all values that satisfy the inequality.

Linear Equations and Inequalities in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Solve	Find the value(s) of the unknown	”Solve 4x − 3 = 2x + 9.”
Solve the inequality	Find range; may need number line	”Solve the inequality 3x + 2 < 14.”
Show your working	Method marks for each step	Required on most equation questions
Write down the inequality	Form inequality from words	”Write an inequality for x if x is at least 5.”
Hence	Use a previous result	”Hence write down the smallest integer value of x.”

Worked exam-style stems (how to answer the wording)

1. “Solve 5(x − 2) = 3x + 4.” Expand: 5x − 10 = 3x + 4. Subtract 3x: 2x − 10 = 4. Add 10: 2x = 14. x = 7. Reward: bracket expanded; terms collected correctly.
2. “Solve the inequality 2 − 3x ≥ 8.” Subtract 2: −3x ≥ 6. Divide by −3, flip sign: x ≤ −2. Reward: sign reversal stated or shown.
3. “List the integers that satisfy −2 < n ≤ 3.” Integers: −1, 0, 1, 2, 3. Reward: −2 excluded (open); 3 included (closed).

Work similar stems on the Algebra topical past paper questions and the Linear Equations and Inequalities quiz.

How Linear Equations connect to the rest of Algebra

Simplifying from Simplifying Algebraic Expressions is the first step in many equation questions. Factorising from Factorisation leads to solving by zero product. Rearranging links to Changing the Subject of the Formula. Use the Cambridge IGCSE Maths resource hub for the full unit.

Common mistakes students make

• Sign errors when moving terms across the equals sign.
• Forgetting to reverse the inequality when dividing by a negative.
• Using a closed circle on the number line for strict < or > (should be open).
• Including boundary values incorrectly in integer-list questions.

When you need more support

If inequality sign reversal or fraction equations keep failing, work through the Changing the Subject quiz and the Algebra topical past paper questions, then get help from a Cambridge IGCSE Maths tutor.

Frequently asked questions

What is the difference between an equation and an inequality? An equation gives an exact value for the unknown. An inequality gives a range — all values that make the statement true.

When do I flip the inequality sign? Only when you multiply or divide both sides by a negative number.

Do I need to draw a number line? Draw one when the question asks you to show the solution on a number line or sketch the values of x.

How do I revise Linear Equations and Inequalities effectively? Solve ten equations and five inequalities including at least two with negative coefficients, then take the Linear Equations and Inequalities quiz.

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