IGCSE Linear Equations and Inequalities: Complete Guide for Cambridge IGCSE Mathematics

IGCSE linear equations and inequalities are fundamental algebra topics in Cambridge IGCSE Mathematics that appear in both Paper 2 and Paper 4. Mastering solving linear equations, linear inequalities, and word problems is essential for achieving top grades in IGCSE exams.

This comprehensive IGCSE linear equations and inequalities guide covers everything you need to know, including solving one-step and multi-step equations, solving inequalities, representing solutions on number lines, worked examples, common exam questions, and expert tips from Tutopiya’s IGCSE maths tutors. We’ll also show you how to avoid the most common mistakes that cost students valuable marks.

🎯 What you’ll learn: By the end of this guide, you’ll know how to solve linear equations, solve and graph inequalities, handle word problems, and apply these skills to solve problems in IGCSE exams.

Already studying with Tutopiya? Practice these skills with our dedicated IGCSE Algebra practice deck featuring exam-style questions and instant feedback.

---

Why IGCSE Linear Equations and Inequalities Matter

IGCSE linear equations and inequalities are essential algebra topics. Here’s why they’re so important:

• High frequency topic: Linear equation questions appear in almost every IGCSE maths paper
• Foundation skill: Essential for solving more complex equations and word problems
• Exam weight: Typically worth 4-8 marks per paper
• Real-world applications: Used in problem-solving across all areas of mathematics
• Problem-solving skills: Develops logical thinking and algebraic manipulation

Key insight from examiners: Students often make errors with signs, forget to check their answers, or struggle with inequalities. This guide will help you master these systematically.

---

Understanding Linear Equations

A linear equation is an equation where the highest power of the variable is 1.

Examples:

• x + 5 = 12
• 3x - 7 = 2x + 4
• 2(x + 3) = 10

---

Solving Linear Equations

One-Step Equations

Example 1: Solve x + 7 = 15

Solution: Subtract 7 from both sides: x + 7 - 7 = 15 - 7 x = 8

Answer: x = 8

Example 2: Solve 3x = 18

Solution: Divide both sides by 3: 3x / 3 = 18 / 3 x = 6

Answer: x = 6

---

Two-Step Equations

Example 3: Solve 2x + 5 = 13

Solution: Step 1: Subtract 5 from both sides 2x + 5 - 5 = 13 - 5 2x = 8

Step 2: Divide both sides by 2 2x / 2 = 8 / 2 x = 4

Answer: x = 4

---

Equations with Brackets

Example 4: Solve 3(x - 2) = 15

Solution: Step 1: Expand brackets 3x - 6 = 15

Step 2: Add 6 to both sides 3x = 21

Step 3: Divide by 3 x = 7

Answer: x = 7

---

Equations with Variables on Both Sides

Example 5: Solve 5x - 3 = 2x + 9

Solution: Step 1: Subtract 2x from both sides 5x - 3 - 2x = 2x + 9 - 2x 3x - 3 = 9

Step 2: Add 3 to both sides 3x = 12

Step 3: Divide by 3 x = 4

Answer: x = 4

---

Understanding Linear Inequalities

Linear inequalities use symbols: <, >, ≤, ≥

Key rule: When multiplying or dividing by a negative number, reverse the inequality sign!

---

Solving Linear Inequalities

Example 6: Solve x + 5 > 12

Solution: Subtract 5 from both sides: x > 7

Answer: x > 7

Example 7: Solve -3x ≤ 15

Solution: Divide both sides by -3 (reverse the sign): x ≥ -5

Answer: x ≥ -5

---

Representing Solutions on Number Lines

• Open circle (○) for < or >
• Closed circle (●) for ≤ or ≥
• Arrow pointing in the direction of the solution

---

Step-by-Step Method

1. Simplify both sides - Expand brackets, collect like terms
2. Move variables - Get all variables to one side
3. Move constants - Get all constants to the other side
4. Solve - Divide or multiply to find the variable
5. Check - Substitute back to verify

---

Common Examiner Traps

• Sign errors - Watch positive/negative signs carefully
• Inequality sign reversal - Reverse sign when multiplying/dividing by negative
• Forgetting to check - Always verify your answer
• Bracket errors - Expand brackets correctly

---

Practice Questions

Question 1

Solve 4x - 7 = 13

Solution: 4x = 20 x = 5

Answer: x = 5

Question 2

Solve 2x + 3 ≥ 11

Solution: 2x ≥ 8 x ≥ 4

Answer: x ≥ 4

---

Tutopiya Advantage: Personalised IGCSE Linear Equations and Inequalities Coaching

• Live whiteboard walkthroughs of equation solving
• Exam-docket homework packs mirroring CAIE specimen papers
• Analytics dashboard so parents see accuracy by topic
• Flexible slots with ex-Cambridge markers for last-mile polishing

📞 Ready to turn shaky equation-solving skills into exam-ready confidence? Book a free IGCSE maths trial and accelerate your revision plan.

---

Frequently Asked Questions About IGCSE Linear Equations and Inequalities

What is a linear equation?

A linear equation is an equation where the highest power of the variable is 1.

How do I solve linear equations?

Use inverse operations to isolate the variable: add/subtract to move constants, multiply/divide to solve.

What’s the difference between equations and inequalities?

Equations use =; inequalities use <, >, ≤, ≥ and represent ranges of values.

When do I reverse the inequality sign?

Reverse the sign when multiplying or dividing both sides by a negative number.

---

Related IGCSE Maths Resources

Strengthen your IGCSE Mathematics preparation with these comprehensive guides:

• IGCSE Quadratic Equations: Complete Guide - Master quadratic equations
• IGCSE Simultaneous Equations: Complete Guide - Master solving simultaneous equations
• IGCSE Maths Revision Notes, Syllabus and Preparation Tips - Complete syllabus overview, topic breakdown, and revision strategies
• IGCSE Past Papers Guide - Access free IGCSE past papers and exam resources

---

Next Steps: Master IGCSE Linear Equations and Inequalities with Tutopiya

Ready to excel in IGCSE linear equations and inequalities? Our expert IGCSE maths tutors provide:

• Personalized 1-on-1 tutoring tailored to your learning pace
• Exam-focused practice with real Cambridge IGCSE past papers
• Interactive whiteboard sessions for visual learning
• Progress tracking to identify and strengthen weak areas
• Flexible scheduling to fit your revision timetable

Book a free IGCSE maths trial lesson and get personalized support to master linear equations and achieve your target grade.

---