Equations, inequalities & graphs

Overview

Solving equations and inequalities, including modulus and higher-order graphs.

Quadratic inequalities

1. Solve the equation = 0 to find the critical values.
2. Sketch the parabola.
3. Read off the region: for > 0 take the parts above the x-axis; for

< 0 the part below.

Modulus equations

To solve |f(x)| = k, solve both f(x) = k and f(x) = −k. Always check your answers in the original equation.

Graphs

• Cubics (y = ax³ + …) have up to two turning points and can cross the x-axis up

to three times.

• Solve equations graphically by finding where two graphs intersect.

Worked example

x² − x − 6 > 0: factorises to (x − 3)(x + 2) > 0 → critical values 3 and −2 → solution x < −2 or x > 3.

Common mistakes

• Writing the inequality the wrong way round — sketch the parabola to be sure.

Exam tips

• For "≤", remember to include the critical values in the solution.