Solving inequalities Revision Notes for Edexcel GCSE Maths

Solving Inequalities

What are Inequalities?

Inequalities are mathematical statements that compare two expressions using symbols like $<$ , $>$ , $≤$ , or $≥$ . Unlike equations, inequalities show that one side is greater than, less than, or equal to the other side.

Inequalities are fundamental tools in mathematics that allow us to express ranges of values rather than exact solutions. They appear frequently in real-world problems where we need to find all possible solutions within certain constraints.

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While equations have one specific solution (or a finite set of solutions), inequalities typically have infinitely many solutions that form a range or interval of values.

Basic Solving Method

Inequalities can be solved using the same steps as equations. You perform the same operations to both sides to isolate the variable.

The key principle is maintaining balance - whatever operation you perform on one side of the inequality, you must perform the exact same operation on the other side.

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Worked Example: Basic Inequality Solving

Solve: $2x - 3 ≤ 15$

Step 1: Add 3 to both sides $2x - 3 + 3 ≤ 15 + 3$ $2x ≤ 18$

Step 2: Divide both sides by 2 $\frac{2x}{2} ≤ \frac{18}{2}$ $x ≤ 9$

Key Rules When Solving Inequalities

The Negative Number Rule

When working with inequalities, there's one crucial rule that distinguishes them from equations:

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Critical Rule: The Sign Flip

When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

This is the most important rule to remember when solving inequalities. Many students forget this step and get incorrect answers.

It's often easier to avoid this complication by rearranging terms to eliminate negative coefficients when possible.

Consider this example: If you have $-2x > 6$ , when you divide both sides by $-2$ , the inequality becomes $x < -3$ . Notice how the sign flipped from $>$ to $<$ .

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Why does the sign flip?

Think about it logically: if $-2 > -4$ (which is true), but when we divide both sides by $-1$ , we get $2 < 4$ (still true). The relationship between the numbers changes when we multiply by negative values.

Solutions to Inequalities

The solution to an inequality has the variable isolated on one side and a number (or expression) on the other side.

Examples of correct solutions:

$x > 4$
$-2 < x$
$x ≤ -\frac{3}{4}$

Examples that are NOT solutions:

$x ≥ 20 + 3$ (number not simplified)
$2x < 10$ (variable not isolated)
$x = 4$ (this is an equation, not an inequality)

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Remember that solutions to inequalities represent ranges of values, not single numbers. Your final answer should clearly show which values the variable can take.

Worked Examples

Example 1: Single Inequalities

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Worked Example: Single Inequalities

(a) Solve: $2x + 1 < 10$

Step 1: Subtract 1 from both sides $2x + 1 - 1 < 10 - 1$ $2x < 9$

Step 2: Divide both sides by 2 $x < \frac{9}{2} = 4\frac{1}{2}$

(b) Solve: $5x ≤ 2x - 6$

Step 1: Subtract $2x$ from both sides $5x - 2x ≤ 2x - 6 - 2x$ $3x ≤ -6$

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

(c) Solve: $12 - 4x ≥ 10$

Step 1: Subtract 12 from both sides $12 - 4x - 12 ≥ 10 - 12$ $-4x ≥ -2$

Step 2: Divide both sides by $-4$ (remember to flip the sign!) $x ≤ \frac{1}{2}$

Example 2: Double Inequalities

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Worked Example: Double Inequalities

Find all integers that satisfy: $-15 ≤ 5n < 10$

This is really two inequalities combined:

$-15 ≤ 5n$ AND $5n < 10$

Solve each part separately:

Part 1: $-15 ≤ 5n$ Divide by 5: $-3 ≤ n$

Part 2: $5n < 10$ Divide by 5: $n < 2$

Combining both conditions: $-3 ≤ n < 2$

The integers that satisfy this are: $n = -3, -2, -1, 0, 1$

Exam Tips

Here are some essential strategies for success when solving inequalities:

Write down the operation you're using at each step for clarity
Remember to apply the same operation to both sides
Your answer can involve negative numbers or fractions - don't be afraid of them
When finding integer solutions, list all whole numbers that satisfy the inequality
Always check that your final answer makes logical sense

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Quick Check Method

After solving, pick a value from your solution range and substitute it back into the original inequality to verify it works. This catches many common errors!

Remember!

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Key Points to Remember:

Solve inequalities exactly like equations - same operations to both sides
Reverse the inequality sign when multiplying or dividing by a negative number
Solutions show a range of values, not just one number
The variable should be isolated on one side of the inequality
Double inequalities can be split and solved separately
Always double-check your work by testing a value from your solution

Solving inequalities (Edexcel GCSE Maths): Revision Notes