IGCSE Mathematics: Linear Equations and Inequalities Practice Questions
To solve a linear equation, do the same thing to both sides until the letter is alone. Inequalities are solved the same way, except that multiplying or dividing by a negative number reverses the inequality sign.
Topic 2.5 of Cambridge IGCSE Mathematics 0580 is examined on both papers and inside word problems throughout. The negative coefficient rule for inequalities is the single most commonly forgotten step. The questions below test it directly.
What you need to know for Linear Equations and Inequalities
- Solving an equationApply the same operation to both sides until the unknown stands alone. Expand brackets and clear fractions first.
- Clearing fractionsMultiply every term on both sides by the lowest common denominator, including any term that is not a fraction.
- Inequality symbolsLess than and greater than are strict. Less than or equal to and greater than or equal to include the endpoint.
- The negative ruleMultiplying or dividing both sides of an inequality by a negative number reverses the direction of the sign.
- Number line notationAn open circle marks a strict inequality where the endpoint is excluded. A filled circle marks an inclusive inequality where it is included.
- CheckingSubstitute your solution back into the original equation. For an inequality, test one value inside the solution range.
IGCSE Mathematics Linear Equations and Inequalities questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Solve 5(x minus 2) = 3x + 8.
Show the worked answer
- Expand the bracket on the left: 5x minus 10 = 3x + 8.
- Subtract 3x from both sides to collect the letters on one side: 2x minus 10 = 8.
- Add 10 to both sides: 2x = 18.
- Divide both sides by 2: x = 9. Check: 5(9 minus 2) = 35, and 3(9) + 8 = 35.
Solve (2x + 1) divided by 3, minus (x minus 4) divided by 2, equals 1.
Show the worked answer
- The lowest common denominator of 3 and 2 is 6. Multiply every term on both sides by 6.
- This gives 2(2x + 1) minus 3(x minus 4) = 6.
- Expand carefully, applying the minus to both terms of the second bracket: 4x + 2 minus 3x + 12 = 6.
- Simplify: x + 14 = 6, so x = minus 8. Check: (minus 16 + 1) divided by 3 = minus 5, and (minus 8 minus 4) divided by 2 = minus 6, and minus 5 minus (minus 6) = 1.
Solve the inequality 3x minus 7 is less than or equal to 11, and describe how the solution is shown on a number line.
Show the worked answer
- Add 7 to both sides: 3x is less than or equal to 18.
- Divide both sides by 3, which is positive, so the sign does not change: x is less than or equal to 6.
- On a number line, mark 6 with a filled circle because the inequality includes 6.
- Draw an arrow from that circle pointing left, showing that every value below 6 is also a solution.
Solve the inequality minus 2x is greater than 8, and explain the step that many students get wrong.
Show the worked answer
- Divide both sides by minus 2 to isolate x.
- Dividing an inequality by a negative number reverses the direction of the sign, so greater than becomes less than.
- minus 2x divided by minus 2 = x, and 8 divided by minus 2 = minus 4, giving x is less than minus 4.
- Check with a test value: x = minus 5 gives minus 2 times minus 5 = 10, which is indeed greater than 8. A value such as x = 0 gives 0, which is not, confirming the direction is right.
Common mistakes in this topic
- Failing to reverse the inequality sign after dividing by a negative.
- Expanding a bracket by multiplying only the first term.
- Sign errors when a bracket is subtracted.
- Using an open circle for an inclusive inequality.
- Forgetting to multiply the non-fraction terms when clearing denominators.
Exam tips
- Solve an inequality exactly like an equation, then check whether you ever divided by a negative.
- Test one value from your solution range in the original inequality. It confirms the direction instantly.
- Multiply every term by the lowest common denominator, including whole numbers on the right hand side.
- Write brackets around any numerator before multiplying through.
- Substitute your answer back into the original equation. It costs seconds and catches sign slips.
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Linear Equations and Inequalities FAQs
When do I reverse an inequality sign?
Only when you multiply or divide both sides by a negative number. Adding or subtracting anything, and multiplying or dividing by a positive number, all leave the direction unchanged. Testing a value from your final range confirms whether you got it right.
How do I solve an equation with fractions?
Multiply every term on both sides by the lowest common denominator, which clears all the fractions in one step. Put brackets around each numerator before multiplying, then expand carefully, particularly where a bracket is being subtracted.
What is the difference between an open and a filled circle on a number line?
An open circle means the endpoint is not included, used for strict inequalities such as less than or greater than. A filled circle means the endpoint is included, used for less than or equal to and greater than or equal to.
How do I check my solution to an equation?
Substitute the value back into the original equation, before any expanding or rearranging, and check that both sides give the same number. If they do not, the error is somewhere in your working rather than in the final arithmetic.
This page covers solving a single equation or inequality. For a pair of equations solved together, see Simultaneous Equations.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.