Learn basic inequalities
An inequality compares values using <, ≤, >, or ≥. Its solution is usually a range of numbers rather than one value.
You will solve a one-variable inequality and know when its sign must reverse.
The idea in plain English
An inequality compares values instead of saying they are exactly equal. Its answer is usually a whole range of numbers.
You solve it like an equation, with one exception: multiplying or dividing by a negative reverses the comparison.
How to recognise this kind of question
- You see <, >, ≤, or ≥ instead of =.
- A variable is being compared with a value.
- The answer describes many possible numbers.
Quick decision: Solve normally; flip the sign only when multiplying or dividing both sides by a negative.
The method
- Use inverse operations to isolate the variable.
- Reverse the inequality sign after multiplying or dividing by a negative number.
- Test a value from the solution range in the original inequality.
Worked examples
−2x < 10
- Divide both sides by −2.
- Reverse < to > because the divisor is negative.
Answer: x > −5
−3x > 12
- Divide both sides by −3.
- Reverse > to < because you divided by a negative.
Answer: x < −4
Your turn—with help
2x + 5 ≤ 13
Need a hint?
Subtract 5, then divide by positive 2.
Show the answer and steps
- Subtract 5: 2x ≤ 8.
- Divide by 2: x ≤ 4.
- The sign stays because 2 is positive.
Answer: x ≤ 4
Watch out for this mistake
Incorrect: −2x > 6, so x > −3
Why it fails: The inequality sign was not reversed after dividing by a negative.
Do this instead: The correct result is x < −3.
Quick check
Try these without looking. Then open each answer to check your thinking.
x − 6 > 2
Need a hint?
Add 6 to both sides.
Show the answer and steps
- x > 2 + 6.
Answer: x > 8
4x ≤ 20
Need a hint?
Divide by positive 4.
Show the answer and steps
- x ≤ 20 ÷ 4.
Answer: x ≤ 5
−2x ≥ 10
Need a hint?
Dividing by −2 reverses the sign.
Show the answer and steps
- Divide both sides by −2.
- Change ≥ to ≤.
Answer: x ≤ −5
Common mistakes
- Forgetting to reverse the sign after dividing by a negative.
- Reversing the sign after adding or subtracting a number.
- Giving one number instead of describing the solution range.
Practice tips
- Circle any negative number used to multiply or divide.
- Check one value that should work and one that should not.