Inequalities
The Skill
Inequality Symbols
| Symbol | Meaning |
|---|---|
| $<$ | Less than |
| $>$ | Greater than |
| $\leq$ | Less than or equal to |
| $\geq$ | Greater than or equal to |
Solving Linear Inequalities
Solve inequalities like equations, but with one important exception:
When you multiply or divide by a negative number, reverse the inequality sign.
Example: Solve $3x + 5 < 17$ $$3x < 12$$ $$x < 4$$
Example with negative: Solve $-2x > 6$ $$x < -3$$ (sign reversed!)
Representing on a Number Line
- Open circle ○ means the value is NOT included ($<$ or $>$)
- Closed circle ● means the value IS included ($\leq$ or $\geq$)
For $x < 4$: open circle at 4, arrow pointing left For $x \geq -2$: closed circle at -2, arrow pointing right
Double Inequalities
$-3 \leq x < 5$ means $x$ is at least $-3$ AND less than 5.
To solve: work with all three parts together, or split into two inequalities.
Example: Solve $7 < 2x + 1 \leq 15$
Subtract 1 from all parts: $$6 < 2x \leq 14$$
Divide all parts by 2: $$3 < x \leq 7$$
Integer Solutions
If asked for integers (whole numbers) in a range:
For $-2 < x \leq 3$, the integers are: $-1, 0, 1, 2, 3$ (Note: $-2$ is not included because of $<$)
The Traps
Common misconceptions and how to avoid them.
Including or excluding wrong integers at boundaries "The Boundary Blunder"
The Mistake in Action
List the integers satisfying $-2 < x \leq 3$
Wrong: $-2, -1, 0, 1, 2, 3$
Why It Happens
Students include -2 even though the $<$ symbol means it's not included, or they may miss that 3 is included because of $\leq$.
The Fix
Check each boundary carefully:
- $-2 < x$ means $x$ is greater than $-2$, so $-2$ is NOT included
- $x \leq 3$ means $x$ is less than or equal to 3, so 3 IS included
Integers in $-2 < x \leq 3$: $-1, 0, 1, 2, 3$
(5 integers, not 6)
Spot the Mistake
List integers in $-2 < x \leq 3$
$-2, -1, 0, 1, 2, 3$
Click on the line that contains the error.
Not reversing the sign when dividing by negative "The Flip Forget"
The Mistake in Action
Solve $-3x < 12$
Wrong: $x < -4$
Why It Happens
Students divide by $-3$ correctly but forget to reverse the inequality sign. They treat it exactly like an equation.
The Fix
When you multiply or divide by a negative number, you must reverse the inequality sign.
$-3x < 12$
Divide by $-3$ AND flip the sign: $$x > -4$$
Why? If $-3x < 12$, think about $x = -5$: $-3(-5) = 15$, and $15 < 12$ is FALSE. But $x = -3$: $-3(-3) = 9$, and $9 < 12$ is TRUE. So $x$ must be greater than $-4$, not less than.
Spot the Mistake
Solve $-3x < 12$
Divide by $-3$: $x < -4$
Click on the line that contains the error.
Using wrong circle type on number line "The Circle Confusion"
The Mistake in Action
Represent $x \leq 3$ on a number line.
Wrong: Open circle at 3, arrow pointing left.
Why It Happens
Students mix up when to use open vs closed circles, or may not have learned the distinction clearly.
The Fix
Open circle ○: Value NOT included ($<$ or $>$) Closed circle ●: Value IS included ($\leq$ or $\geq$)
For $x \leq 3$:
- The symbol is $\leq$ (less than or equal to)
- 3 IS included
- Use a closed (filled) circle ●
Memory aid: "Equal means fill-ed" (the line under the symbol means fill in the circle)
Spot the Mistake
Represent $x \leq 3$
Open circle at 3
Click on the line that contains the error.
Writing the inequality the wrong way round "The Backwards Inequality"
The Mistake in Action
Solve $5 > 2x - 1$
Wrong: $6 > 2x$ $3 > x$
Therefore $x > 3$
Why It Happens
Students solve correctly but then flip the answer when writing it in the conventional form, not realizing that $3 > x$ and $x > 3$ mean completely different things.
The Fix
$3 > x$ and $x > 3$ are opposites!
- $3 > x$ means "3 is greater than $x$" → same as $x < 3$
- $x > 3$ means "$x$ is greater than 3"
Your working gives $3 > x$, which means $x < 3$.
Tip: Read the inequality aloud. "3 is greater than $x$" means $x$ must be small values (less than 3).
Spot the Mistake
$5 > 2x - 1$
$6 > 2x$, then $3 > x$
Therefore $x > 3$
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
Solve $4x + 3 > 19$
Solution
Step 1: Subtract 3 from both sides. $$4x > 16$$
Step 2: Divide both sides by 4. $$x > 4$$
Answer: $x > 4$
Check: Try $x = 5$: $4(5) + 3 = 23$, and $23 > 19$ ✓ Try $x = 4$: $4(4) + 3 = 19$, and $19 > 19$ is false ✓ (4 is not in the solution)
Question
Solve $7 - 2x \geq 13$
Solution
Step 1: Subtract 7 from both sides. $$-2x \geq 6$$
Step 2: Divide both sides by $-2$. Remember: Dividing by negative → reverse the sign! $$x \leq -3$$
Answer: $x \leq -3$
Check: Try $x = -4$: $7 - 2(-4) = 7 + 8 = 15$, and $15 \geq 13$ ✓ Try $x = -2$: $7 - 2(-2) = 7 + 4 = 11$, and $11 \geq 13$ is false ✓
Question
Solve $-5 < 3x + 1 \leq 16$
Solution
Work with all three parts at once.
Step 1: Subtract 1 from all parts. $$-5 - 1 < 3x + 1 - 1 \leq 16 - 1$$ $$-6 < 3x \leq 15$$
Step 2: Divide all parts by 3. $$\frac{-6}{3} < \frac{3x}{3} \leq \frac{15}{3}$$ $$-2 < x \leq 5$$
Answer: $-2 < x \leq 5$
This means $x$ is greater than $-2$ AND less than or equal to 5.
Question
Find all integers $n$ such that $-3 \leq 2n - 1 < 7$
Solution
Step 1: Solve the inequality.
Add 1 to all parts: $$-2 \leq 2n < 8$$
Divide all parts by 2: $$-1 \leq n < 4$$
Step 2: List the integers. $n$ is at least $-1$ (included because $\leq$) $n$ is less than 4 (not included because $<$)
Integers: $-1, 0, 1, 2, 3$
Answer: $n = -1, 0, 1, 2, 3$ (5 integers)
Level 2: Scaffolded
Fill in the key steps.
Question
Represent $x \geq -2$ on a number line.
Level 3: Solo
Try it yourself!
Question
A taxi charges £3 plus £2 per mile. Tom has £15. Write an inequality for the number of miles $m$ he can travel, and solve it.
Show Solution
Set up the inequality: Total cost = $3 + 2m$ Tom has £15, so cost must be at most £15: $$3 + 2m \leq 15$$
Solve: $$2m \leq 12$$ $$m \leq 6$$
Answer: Tom can travel at most 6 miles.
Check: 6 miles costs $3 + 2(6) = £15$ ✓ 7 miles would cost $3 + 2(7) = £17$, which is too much ✓
Examiner's View
Mark allocation: Solving: 2 marks. Number line: 1 mark. Integer solutions: 1 mark.
Common errors examiners see:
- Forgetting to flip the sign when dividing by negative
- Open/closed circle confusion
- Missing integers at the boundaries
- Writing the answer the wrong way round
What gains marks:
- Clear number line with correct circles
- Listing integers correctly
- Showing working for each step
AQA Notes
AQA often asks for solutions on a number line AND as integers.