Solving inequalities Revision Notes for AQA GCSE Maths

Solving inequalities

What are inequalities?

Inequalities are mathematical statements that show the relationship between two expressions using symbols like $<$ , $>$ , $≤$ , or $≥$ . Unlike equations which have an equals sign, inequalities tell us that one side is greater than, less than, greater than or equal to, or less than or equal to the other side.

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Inequalities are everywhere in real life! Think about speed limits (your speed ≤ 70 mph), age restrictions (you must be ≥ 18), or temperature ranges (keep medicine between 2°C and 8°C).

Basic method for solving inequalities

The good news is that inequalities can be solved using exactly the same steps as equations. You can add, subtract, multiply, and divide both sides of an inequality just like you would with an equation. The goal is to get the letter (variable) on its own on one side.

For example, if you have $2x - 3 = 15$ , you would:

Add 3 to both sides: $2x = 18$
Divide both sides by 2: $x = 9$

The same steps work for inequalities like $2x - 3 < 15$ .

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This similarity to equations makes inequalities much less intimidating once you realise the process is almost identical!

The crucial rule about negative numbers

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Watch out! There is one very important difference when solving inequalities. If you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign.

This means:

$<$ becomes $>$
$>$ becomes $<$
$≤$ becomes $≥$
$≥$ becomes $≤$

Top tip: It's usually easier to avoid negative numbers altogether by adding terms to both sides instead of subtracting them.

What do solutions look like?

Solutions to inequalities have the variable on its own on one side and a number on the other side. Unlike equations which usually have one specific answer, inequalities often have many possible solutions.

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Examples of correct solutions:

$x > 4$ (x is greater than 4)
$-2 < x$ (x is greater than -2)
$x ≤ -\frac{3}{4}$ (x is less than or equal to -3/4)

Examples that are NOT solutions:

$x ≥ 20 + 3$ (not simplified)
$2x < 10$ (x is not on its own)
$x = 4$ (this is an equation, not an inequality)

Step-by-step method

Write down the operation you're using at each step
Remember to do the same thing to both sides
Check if you're multiplying or dividing by a negative number
Reverse the inequality sign if needed
Simplify to get the variable on its own

Worked examples

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Worked Example 1: Simple inequality

Solve $2x + 1 < 10$

Step 1: Subtract 1 from both sides $2x < 9$

Step 2: Divide both sides by 2 $x < 4.5$

Solution: $x < 4.5$

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Worked Example 2: Inequality with negative coefficient

Solve $5x ≤ 2x - 6$

Step 1: Subtract 2x from both sides $3x ≤ -6$

Step 2: Divide both sides by 3 $x ≤ -2$

Solution: $x ≤ -2$

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Worked Example 3: Reversing the sign

Solve $12 - 4x ≥ 10$

Step 1: Add 4x to both sides $12 ≥ 10 + 4x$

Step 2: Subtract 10 from both sides $2 ≥ 4x$

Step 3: Divide both sides by 4 (positive number, so no sign change needed) $\frac{1}{2} ≥ x$ or $x ≤ \frac{1}{2}$

Solution: $x ≤ \frac{1}{2}$

Double inequalities

Sometimes you might see inequalities like $-15 ≤ 5n < 10$ . This is really two inequalities combined:

$-15 ≤ 5n$ AND $5n < 10$

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Solving double inequalities:

Solve them separately:

$-15 ≤ 5n$ gives $n ≥ -3$
$5n < 10$ gives $n < 2$

So the solution is $-3 ≤ n < 2$ , which means n can be any value from -3 (including -3) up to but not including 2.

Exam tips

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Essential exam strategies:

Write down each operation you perform - this shows your working clearly
Don't use equals signs in your final answer unless specifically asked
Your answer can involve negative numbers or fractions - this is perfectly normal
Check your answer by substituting a value back into the original inequality

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Key Points to Remember:

Inequalities are solved exactly like equations, using the same operations on both sides
Always reverse the inequality sign when multiplying or dividing by a negative number
Solutions have the variable alone on one side and a number on the other
Avoid negative coefficients when possible by adding terms instead of subtracting
Double inequalities are really two separate inequalities that need solving individually

Solving inequalities (AQA GCSE Maths): Revision Notes