Rukasha Maths

GCSE maths Beta

Linear equations and inequalities · Revision guide

Foundation · Algebra · 1 page in the PDF

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Learning objectives

  • Solve a linear equation using balanced operations.
  • Solve and represent a linear inequality.

Key ideas

Apply the same operation to both sides of an equation; check by substituting the solution.

Expand brackets, collect variable terms on one side, then isolate the unknown.

Strict inequalities exclude the boundary; inclusive inequalities include it.

<, > exclude;≤, ≥ include<,\ >\ \text{exclude};\qquad\le,\ \ge\ \text{include}

Multiplying or dividing an inequality by a negative reverses the inequality sign.

On a number line: an open circle excludes the boundary; a filled circle includes it.

Worked examples

Example 1

Solve the equation by keeping both sides balanced.

3(x+2)=2x+11⟹3x+6=2x+113(x+2)=2x+11\quad\Longrightarrow\quad3x+6=2x+11
x+6=11⟹x=5x+6=11\quad\Longrightarrow\quad x=5

Check both sides.

3(5+2)=21=2(5)+113(5+2)=21=2(5)+11
x=5x=5

Example 2

Solve the inequality, reversing the sign when dividing by a negative.

−2x<8⟹x>−4-2x<8\quad\Longrightarrow\quad x>-4
x>−4x>-4

Watch out

Reverse the inequality sign only when multiplying or dividing by a negative, not when adding or subtracting.