Linear Inequalities (OCR AS Maths: Pure)

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Linear Inequalities

What are linear inequalities?

  • Linear inequalities are similar to equations but answers take a range of values
  • Linear means there will be no terms other than degree 1
    • no squared terms or higher powers, no fractional or negative powers
  • Inequalities use the symbols following symbols
    • greater thanGreater than e.g. 5 space greater than space 3
    • less thanLess than e.g. negative 8 space less than space 7
    • greater or equal than Greater than or equal to
    • less or equal thanLess than or equal to
  • Inequalities can be represented in many ways using number lines, set notation and interval notation

2.4.1 Linear Inequalities Notes Diagram 4, Edexcel A Level Maths: Pure revision notes

Number line diagrams 

  • Number line diagrams are made up from circles and lines set above a number line
    • A filled-in circle or empty circle above a number denotes whether the number is included or not
      • filled in for the greater/less than or equal to symbolsless or equal than space greater or equal than
      • empty for the greater/less than symbolsless than space greater than
    • Arrows show the range of values that are allowed

2.4.1 Linear Inequalities Notes Diagram 1, Edexcel A Level Maths: Pure revision notes

Set notation

  • Set notation is a formal way of writing a range of values
  • Use of curly brackets { }
  • Intersection ∩ and union ∪ may be used
  • Not to be confused with interval notation

2.4.1 Linear Inequalities Notes Diagram 2, Edexcel A Level Maths: Pure revision notes

Interval notation

  • Interval notation uses different brackets to indicate whether a number is included or not
  • Use of square [] and round () brackets
  • [ or ] mean included
  • ( or ) mean excluded
    • (4,8] means 4 < x < 8
  • Note ∞ always uses ( or )
  • Not to be confused with set notation

2.4.1 Linear Inequalities Notes Diagram 3, Edexcel A Level Maths: Pure revision notes 

Skills for solving linear inequalities

  • representing and interpreting inequalities displayed on a number line
  • writing and interpreting set notation
    • eg {x : x > 1} ∩ {x : x ≤ 7} is the same as 1 < x ≤ 7

  • writing and interpreting interval notation
    • eg [-4, 6) is the same as -4 ≤ x < 6

How do I solve linear inequalities?

  • Treat the inequality as an equation and solve
    • avoid multiplying or dividing by a negative
    • if unavoidable, “flip” the inequality sign so < → >, ≥ → ≤, etc
    • try to rearrange to make the x term positive

2.4.1 Linear Inequalities Notes Diagram 5, Edexcel A Level Maths: Pure revision notes

Worked example

2.4.1 Linear Inequalities Example Diagram, Edexcel A Level Maths: Pure revision notes

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.