Inequalities
Inequalities are like equations, but instead of saying two expressions are equal, they compare which side is bigger or smaller.
x > 3 means x is greater than 3.
x < 3 means x is less than 3.
x ≥ 3 means x is greater than or equal to 3.
x ≤ 3 means x is less than or equal to 3.
Solving simple inequalities
You solve inequalities using similar steps to solving equations. The aim is to get the variable on its own.
Example 1: Solve x + 5 > 12
Subtract 5 from both sides:
x > 7
The solution is x > 7.
Example 2: Solve 3x ≤ 18
Divide both sides by 3:
x ≤ 6
The solution is x ≤ 6.
Inequalities with two steps
For two-step inequalities, undo addition or subtraction first, then undo multiplication or division.
Example 3: Solve 2x + 3 < 11
Subtract 3 from both sides:
2x < 8
Divide by 2:
x < 4
Reversing the inequality sign
If you multiply or divide both sides by a negative number, reverse the inequality sign.
Example 4: Solve -2x > 10
Divide both sides by -2.
Because we divide by a negative number, reverse the sign:
x < -5
Showing inequalities on number lines
Inequalities can be shown on a number line using open or closed circles.
Use an open circle for < or >.
Use a closed circle for ≤ or ≥.
Example 5: Number line meaning
x > 4 means values greater than 4, but not including 4.
Use an open circle at 4 and shade to the right.
x ≥ 4 includes 4, so use a closed circle.
Integer solutions
Some GCSE questions ask for integer solutions. Integers are whole numbers, including negative whole numbers and zero.
Example 6: Find integer solutions
Find the integer solutions of 1 < x ≤ 5.
The integers greater than 1 and less than or equal to 5 are:
2, 3, 4, 5
A common mistake is forgetting to reverse the inequality sign when dividing by a negative number. This changes the final answer.
Always read the inequality sign carefully. The difference between < and ≤ can change whether an endpoint is included.
Video explanation
A short Worthing Maths Tutor video explanation for inequalities can be embedded here later to improve student engagement and time on page.
Practice questions
- Solve x + 4 > 10.
- Solve 5x ≤ 20.
- Solve 3x - 2 < 13.
- Solve -4x ≥ 12.
- List the integer solutions of -2 ≤ x < 3.
Answers
- x > 6
- x ≤ 4
- x < 5
- x ≤ -3
- -2, -1, 0, 1, 2
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