Solving linear equations and inequalities

Linear equations

A linear equation contains an unknown (usually x) raised only to the first power. The goal is to isolate x on one side.

The golden rule: whatever you do to one side, do the same to the other.

Types of linear equation:

Type 1 — Simple: 3x + 7 = 16. 3x = 9 (subtract 7 from both sides) x = 3 (divide both sides by 3).

Type 2 — Variables on both sides: 5x − 3 = 2x + 9. 3x = 12 (subtract 2x from both sides; add 3) x = 4.

Type 3 — Brackets: 4(2x − 1) = 3x + 11. 8x − 4 = 3x + 11 (expand brackets) 5x = 15 x = 3.

Type 4 — Fractions: (2x + 3)/4 = 5. 2x + 3 = 20 (multiply both sides by 4) 2x = 17 x = 8.5.

Type 5 — Fractions with variables in denominator: careful — check for solutions that would make the denominator zero.

Forming equations from worded problems

This is where CCEA differs: they often present a context (area of a rectangle, ages, costs) and ask you to form and solve an equation. The process:

1. Define a variable (let x = ...).
2. Write an equation from the context.
3. Solve the equation.
4. Check your answer makes sense in context.

Linear inequalities

Inequalities are solved like equations, with one crucial difference: if you multiply or divide by a negative number, the inequality sign flips.

Example: −2x < 6 → divide both sides by −2, flip sign → x > −3.

Representing on a number line:

• Strict inequality (< or >): open circle at the end.
• Non-strict (≤ or ≥): filled (solid) circle.

Set notation: {x : x > −3} or −3 < x.

Double inequalities: −1 ≤ 3x + 2 < 11. Subtract 2: −3 ≤ 3x < 9. Divide by 3: −1 ≤ x < 3.

Integer solutions from an inequality

"List all integer values of x such that −1 ≤ x < 3" → x = −1, 0, 1, 2 (note: 3 is excluded by strict inequality).