Inequalities are a common topic in GCSE Maths. You need to know how to solve inequalities, represent answers on a number line, and deal with inequalities involving negative numbers and quadratic expressions.
The good news is that solving an inequality is very similar to solving an equation — with one important rule to remember.
Understand the inequality symbols
| Symbol | Meaning | Example |
|---|---|---|
| $<$ | Less than | $x<5$ |
| $>$ | Greater than | $x>5$ |
| $\leq$ | Less than or equal to | $x\leq5$ |
| $\geq$ | Greater than or equal to | $x\geq5$ |
For example, $x<5$ means that $x$ can be any number smaller than 5, such as 4, 0 or -3.
However, $x\leq5$ also allows $x=5$.
How to solve inequalities
Treat an inequality like an equation: collect the $x$ terms on one side and the numbers on the other.
Example
Solve $x+3<8$
Step 1: Subtract 3 from both sides.
\[ x+3-3<8-3 \]
Step 2: Simplify.
\[ \boxed{x<5} \]
Example
Solve $3x+2\leq14$
Step 1: Subtract 2 from both sides.
\[ 3x\leq12 \]
Step 2: Divide both sides by 3.
\[ \boxed{x\leq4} \]
Example
Solve $5x-7>13$
Step 1: Add 7 to both sides.
\[ 5x>20 \]
Step 2: Divide both sides by 5.
\[ \boxed{x>4} \]
The key is to perform the same operation on both sides, just as you would when solving equations.
The most important rule: multiplying or dividing by a negative number
If you multiply or divide both sides by a negative number, you must reverse
$>$ becomes$<$
$\leq$ becomes $\geq$
$\geq$ becomes $\leq$
Example
Solve $-2x<10$
Step 1: Divide both sides by $-2$.
Step 2: Reverse the inequality sign.
\[ x>\frac{10}{-2} \]
Therefore,
\[ \boxed{x>-5} \]
Check: $x=-4$ works because $-2(-4)=8$, and $8<10$.
A common GCSE mistake is to divide by a negative number but forget to reverse the inequality sign.
Representing inequalities on a number line
There are two types of circles to remember.
Example A: $x>2$
Open circle at 2: 2 is not included.
Arrow to the right: values greater than 2.
Example B: $x\leq2$
- Filled circle at 2: 2 is included.
- Arrow to the left: values less than or equal to 2.
Remember:
- Strict inequalities ($<$ and $>$) use an open circle.
- Inclusive inequalities ($\leq$ and $\geq$) use a filled circle.
Solving double inequalities
A double inequality gives you both a lower and an upper limit.
For example:
\[ 2<x+1\leq7 \]
Step 1: Subtract 1 from all three parts.
\[ 2-1<x+1-1\leq7-1 \]
Step 2: Simplify.
\[ \boxed{1<x\leq6} \]
This means \[x\] is greater than 1 but less than or equal to 6.
Writing inequalities from word problems
GCSE questions often describe a real-life situation and ask you to form an inequality.
Example
A cinema ticket costs £8. A student has £40 and wants to buy as many tickets as possible. Write an inequality for the number of tickets $n$.
Step 1: The cost of $n$ tickets is $8n$.
Step 2: The cost cannot exceed £40.
\[ \boxed{8n\leq40} \]
Step 3: Divide by 8.
\[ \boxed{n\leq5} \]
Since you cannot buy a fraction of a ticket, the maximum is 5 tickets.
Look out for these phrases:
| Words in the question | Symbol |
|---|---|
| Greater than | $>$ |
| Less than | $<$ |
| At least | $\geq$ |
| At most | $\leq$ |
| No more than | $\leq$ |
| More than | $>$ |
| Minimum | Usually $\geq$ |
| Maximum | Usually $\leq$ |
Final exam checklist
Before you finish an inequality question, check:
- Have you collected the $x$ terms and numbers correctly?
- If you divided or multiplied by a negative number, did you reverse the sign?
- If you drew a number line, did you use the correct circle and direction?
- If the question involves real life, have you interpreted the answer sensibly?
The single most important rule is to reverse the inequality sign whenever you multiply or divide both sides by a negative number.