Linear Inequalities (AQA GCSE Further Maths): Revision Note

Linear inequalities

What is a linear inequality?

• An inequality tells you that one expression is greater than (“>”) or less than (“<”) another

  • “⩾” means “greater than or equal to”

  • “⩽” means “less than or equal to”

• A linear inequality only has constant terms (numbers with no letters) and terms in x (and/or a y); but no x² terms or terms with higher powers of x

  • For example, 3x² > 12 is not a linear inequality (it is a quadratic inequality)

• For example, 3x + 4 ⩾ 7 would be read “3x + 4 is greater than or equal to 7”. 

How do I solve linear inequalities?

• Solving linear inequalities is just like Solving Linear Equations

  • Follow the same rules, but keep the inequality sign throughout

  • If you change the inequality sign to an equals sign you are changing the meaning of the problem

• When you multiply or divide both sides by a negative number, you must flip the sign of the inequality 

  • e.g. 1 < 2 → [times both sides by (–1)] → –1 > –2 (sign flips)

• Never multiply or divide by a variable (x) as this could be positive or negative

• The safest way to rearrange is simply to add & subtract to move all the terms onto one side

• You also need to know how to deal with “Double” Inequalities

How do I solve double inequalities?

• Inequalities such as $a<2x<b$ can be solved by doing the same thing to all three parts of the inequality

  • Use the same rules as solving linear inequalities