Inequalities (Edexcel GCSE Maths): Revision Notes

Inequalities

What are inequalities?

Inequalities are mathematical statements that compare two values, showing whether one is greater than, less than, or equal to another. Don't worry - once you understand the basic principles, solving inequalities is very similar to solving regular equations!

Understanding inequality symbols

Learning the inequality symbols is your first step to mastering this topic. Each symbol has a specific meaning that tells you how two values relate to each other.

The four main inequality symbols are:

• $>$ means "greater than"
• $\<$ means "less than"
• $\geq$ means "greater than or equal to"
• $\leq$ means "less than or equal to"

Solving inequalities algebraically

The wonderful news about inequalities is that you can solve them using almost exactly the same methods as regular equations. You can add, subtract, multiply, and divide both sides to isolate your variable.

When solving inequalities, follow these familiar steps:

1. Simplify both sides if needed
2. Add or subtract terms to get variables on one side
3. Add or subtract constants to isolate the variable term
4. Multiply or divide to get the variable by itself

The process feels natural because it mirrors equation solving techniques you already know.

The crucial rule about negative numbers

Here's the one critical difference between solving equations and inequalities: when you multiply or divide both sides by a negative number, you must flip the inequality sign.

This might seem tricky at first, but with practice, you'll remember to check whether you're dividing or multiplying by a negative number and flip the sign accordingly.

Representing inequalities on number lines

Visual representation helps you understand inequality solutions better. Number lines show the range of values that satisfy your inequality using specific symbols.

When drawing inequalities on number lines:

• Use an open circle (○) for $>$ or $\<$ symbols (the value is not included)
• Use a closed circle (●) for $\geq$ or $\leq$ symbols (the value is included)
• Draw a line or arrow showing all values that satisfy the inequality

For compound inequalities like $-4 \< x \leq 3$, you're showing all values between -4 and 3, where -4 is not included but 3 is included.

Working with different types of problems

Inequalities appear in various forms, from simple one-step problems to compound inequalities involving multiple conditions.

When dealing with integer solutions, remember that you're looking for whole number values (positive, negative, or zero) that satisfy your inequality. Always check your work by substituting a value back into the original inequality.

For compound inequalities, treat each part of the inequality separately while maintaining the relationship between all parts. This means performing the same operation to all sections of the compound inequality.