Solving linear equations
A linear equation has the unknown to the first power only. To solve, do the same operation to both sides to keep the balance, until the letter is alone. Work in reverse order of operations: undo +/− first, then ×/÷.
Key idea
Whatever you do to one side you must do to the other. To check, substitute your answer back into the original equation.
Worked example
Solve 3(x − 2) = 12.
Expand: 3x − 6 = 12.
Add 6 to both sides: 3x = 18.
Divide both sides by 3: x = 6.
Check: 3(6 − 2) = 3 × 4 = 12. ✓
Solving inequalities
Inequalities (<, >, ≤, ≥) are solved the same way as equations, with one crucial rule.
Remember
- When you multiply or divide both sides by a negative number, reverse the inequality sign.
- x > 4 means all values greater than 4 (4 not included).
- x ≥ 4 includes 4 itself.
Reading a solution
The solution to an inequality is a range of values, often shown on a number line: an open circle (○) for < or >, a filled circle (●) for ≤ or ≥. For example, solving −2x < 6 gives x > −3 (sign reversed on dividing by −2).