Solving like an equation
A linear inequality in one variable is like a linear equation but with an inequality symbol, for example x + 3 > 7 or 2x − 1 ≤ 9. We solve it almost the same way: add, subtract, multiply or divide both sides to get the variable alone.
Worked example / Worked example
Solve x + 4 < 10.
Subtract 4 from both sides: x < 10 − 4.
So x < 6. Any number less than 6 is a solution.
The important rule
There is one special rule that makes inequalities different from equations.
Key idea / Key idea
When you multiply or divide both sides by a negative number, reverse the inequality sign. For example, if −x < 3, multiply by −1 to get x > −3.
Worked example / Worked example
Solve 3x − 2 ≥ 7.
Add 2 to both sides: 3x ≥ 9.
Divide both sides by 3 (positive, so no reversal): x ≥ 3.
Check with x = 4: 3 × 4 − 2 = 10 ≥ 7. Correct.
Remember / Remember
- Solve like an equation to isolate the variable.
- Reverse the sign only when × or ÷ by a negative.
- Check by testing a value in the solution.