Revision NotesExam QuestionsPast Papers
O LevelMathsCambridgeRevision NotesAlgebra & SequencesLinear Equations & InequalitiesSolving Linear Inequalities

Solving Linear Inequalities (Cambridge O Level Maths)

Revision Note

Amber

Author

Amber

Last updated

Solving 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 just has an x (and/or a y) etc in it and no x2 terms or terms with higher powers of x
  • 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 use Number Lines and deal with “Double” Inequalities

How do I represent linear inequalities on a number line?

  • Inequalities such as x space less than space a and x space greater than space a can be represented on a normal number line using an open circle and an arrow
    • For less than, the arrow points to the left of a
    • For greater than, the arrow points to the right of a
  • Inequalities such as x space less or equal than space a and x space greater or equal than space a can be represented on a normal number line using a solid circle and an arrow
    • For less or equal than, the arrow points to the left of a
    • For greater or equal than, the arrow points to the right of a
  • Inequalities such as a space less than space x space less than space b and a space less or equal than space x space less or equal than space b can be represented on a normal number line using two circles at a and b and a line between them
    • For less than or greater than use an open circle
    • For less or equal than or greater or equal than, use a solid circle
  • Disjoint inequalities such as "x less than a or x greater than b" can be represented with two circles at a and b, an arrowed line pointing left from a and an arrowed line pointing right from b, and a blank space between a and b

Solving Inequalities - Linear RN1, downloadable IGCSE & GCSE Maths revision notes

How do I solve double inequalities?

  • Inequalities such as a space less than space 2 x space less than space b can be solved by doing the same thing to all three parts of the inequality
    • Use the same rules as solving linear inequalities

Examiner Tip

  • Do not change the inequality sign to an equals when solving linear inequalities, you will lose marks in an exam for doing this. 

Worked example

(a)
Solve the inequality negative 7 space less or equal than space 3 x space minus space 1 space less than space 2, illustrating your answer on a number line.

This is a double inequality, so any operation carried out to one side must be done to all three parts.
Use the expression in the middle to choose the inverse operations needed to isolate x.
Add 1 to all three parts.
Remember not to change the inequality signs.
negative 6 space less or equal than space 3 x space less than space 3
Divide all three parts by 3.
3 is positive so there is no need to flip the signs.
bold minus bold 2 bold space bold less or equal than bold space bold italic x bold space bold less than bold space bold 1
Illustrate the final answer on a number line, using an open circle at 1 and a closed circle at -2.
2-18-solving-inequalities
(b)
Give your answer to part (a) in set notation

Rewrite your answer using the set notation rules discussed above

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourself

    Did this page help you?

    Previous:Solving Linear EquationsNext:Solving Quadratics by Factorising
    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.