Equations and Inequalities
What this domain tests
Linear equations, simultaneous pairs, simple quadratics and linear inequalities, plus short word problems that must first be translated into one of these forms.
Core rules
- Do the same operation to both sides. Clear fractions by multiplying every term by the LCM of the denominators.
- Simultaneous equations: eliminate when coefficients already line up, substitute when one variable is isolated. Always substitute the pair back into both original equations.
- Quadratics: move everything to one side first, then factorise. If integer factors do not appear, use x = (−b ± √(b²−4ac)) / 2a. A product is zero only when one factor is zero.
- Multiplying or dividing an inequality by a negative number reverses the sign; adding or subtracting never does.
- Show boundaries clearly: open for < and >, filled for ≤ and ≥.
Worked example
Solve 3x − 7 > 5x + 1. Subtract 5x: −2x − 7 > 1. Add 7: −2x > 8. Divide by −2 and reverse: x < −4. Check with x = −5: left side −22, right side −24, and −22 > −24 is true, so the direction is right.
Traps that cost marks
Forgetting to reverse at the division step; dropping a negative sign while transposing; solving x² = 4x by dividing by x, which silently discards x = 0 (factorise to x(x−4) = 0 instead); reporting a solution without substituting it back; ignoring the stated domain, such as a length that must be positive.