How to Solve: Linear Equations and Inequalities

How to Solve: Linear Equations and Inequalities

GCSE / A-Level Maths

Introduction

"Mastering linear equations and inequalities unlocks 10–15% of your GCSE Maths exam marks—and real-life problems like budgeting, engineering, or even deciding how many pizzas to order for a party. One wrong sign, and your answer could be off by 100%—let’s make sure that doesn’t happen."

What You Need To Know First

Before diving in, you must already understand: 1. Basic algebra – How to expand brackets, collect like terms, and simplify expressions. 2. Negative numbers – Rules for multiplying/dividing negatives (e.g., – × – = +). 3. Balancing equations – Whatever you do to one side, you must do to the other.

If any of these feel shaky, pause and review them first.

Key Vocabulary

Formulas To Know

1. Solving Linear Equations

General form: ax + b = c - a = coefficient of x - b = constant term - c = right-hand side (RHS) value MEMORISE THIS: The goal is to isolate x using inverse operations.

2. Solving Linear Inequalities

General form: ax + b > c (or <, ≥, ≤) - Same as equations, but flip the inequality sign when multiplying/dividing by a negative number. MEMORISE THIS: "Negative flip!

3. Number Line Representation (Inequalities)

• Open circle (○) = > or < (value not included).
• Closed circle (●) = ≥ or ≤ (value is included).
• Arrow = All values in that direction.

Step-by-Step Method

Solving Linear Equations

Step 1: Simplify both sides. - Expand brackets. - Collect like terms.

Step 2: Move all x-terms to one side, constants to the other. - Use inverse operations (add/subtract first, then multiply/divide).

Step 3: Isolate x. - Divide both sides by the coefficient of x.

Step 4: Check your answer. - Substitute x back into the original equation.

Solving Linear Inequalities

Step 1: Solve like an equation (follow Steps 1–3 above).

Step 2: If you multiply or divide by a negative number, flip the inequality sign.

Step 3: Write the solution in inequality form (x > 3) or as a number line.

Step 4: Check a value in the solution set to verify.

Worked Examples

Example 1 – Basic Linear Equation

Problem: Solve 3x + 5 = 14.

Step 1: Subtract 5 from both sides. 3x + 5 – 5 = 14 – 5 3x = 9

Step 2: Divide both sides by 3. 3x ÷ 3 = 9 ÷ 3 x = 3

Check: 3(3) + 5 = 9 + 5 = 14 ✔️

What we did and why: - We used inverse operations to isolate x. - Subtraction first (to move the constant), then division (to remove the coefficient).

Example 2 – Medium (Brackets & Negatives)

Problem: Solve 2(4 – x) = 10.

Step 1: Expand the bracket. 8 – 2x = 10

Step 2: Subtract 8 from both sides. –2x = 2

Step 3: Divide by –2 (remember: negative flip if it were an inequality!). x = –1

Check: 2(4 – (–1)) = 2(5) = 10 ✔️

What we did and why: - Expanded first to simplify. - Moved constants, then divided by the coefficient (even if negative).

Example 3 – Exam-Style (Disguised Equation)

Problem: "The sum of three consecutive even numbers is 78. Find the smallest number."

Step 1: Define the variable. - Let the smallest number be x. - Next two even numbers: x + 2 and x + 4.

Step 2: Write the equation. x + (x + 2) + (x + 4) = 78

Step 3: Simplify. 3x + 6 = 78

Step 4: Solve. 3x = 72 x = 24

Check: 24 + 26 + 28 = 78 ✔️

What we did and why: - Translated words into algebra. - Simplified before solving to avoid mistakes.

Example 4 – Inequality (Negative Coefficient)

Problem: Solve –2x + 3 ≤ 7.

Step 1: Subtract 3 from both sides. –2x ≤ 4

Step 2: Divide by –2 (flip the sign!). x ≥ –2

Step 3: Draw the number line. - Closed circle at –2, arrow to the right.

Check: Test x = 0 (should satisfy 0 ≥ –2 ✔️).

What we did and why: - Followed equation steps, but remembered the negative flip rule.

Common Mistakes

Exam Traps

1-Minute Recap

"Here’s the night-before cheat sheet: 1. Equations: Simplify, move x terms to one side, constants to the other, then divide. Always check! 2. Inequalities: Same as equations, but flip the sign if you multiply/divide by a negative. 3. Word problems: Define x, write the equation, solve, and check if it makes sense. 4. Watch out for: Negative coefficients, brackets, and hidden traps like –x. 5. Exam tip: If stuck, plug in answer choices—sometimes that’s faster than solving!

You’ve got this. Now go ace that exam!