Equations, Inequalities and Graphs
Core ideas
Candidates must solve linear and quadratic equations, simultaneous equations (by elimination or substitution), and equations involving algebraic fractions. Inequalities require care with direction: multiplying or dividing by a negative number reverses the sign. Graphical solutions include reading intersections of a curve with a line, and using sketches to determine where f(x) > g(x). Also examinable: solving modulus equations such as |2x − 1| = 5 (split into two cases), and quadratic inequalities, best handled by finding roots then testing regions or sketching the parabola.
Traps
- Failing to flip the inequality sign when multiplying by a negative.
- Dividing a quadratic inequality by x without considering x < 0.
- Forgetting that |expression| = k has two solutions when k > 0.
- Losing solutions when squaring both sides of an equation.
- Not checking answers in the original equation, especially with fractions.
Worked example
Solve x² − x − 6 < 0. Factorise: (x − 3)(x + 2) < 0. The parabola opens upward, so it lies below the axis between the roots. Hence −2 < x < 3. If the inequality were > 0 the answer would be x < −2 or x > 3.
Rule of thumb
For any quadratic inequality, factorise, sketch the parabola, and read off the region that satisfies the sign required - never guess.