Linear Inequalities (AQA AS Maths): Revision Note

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 than e.g. 

  • Less than e.g. 

  • Greater than or equal to

  • Less than or equal to

• Inequalities can be represented in many ways using number lines, set notation and interval notation

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 symbols

    • empty for the greater/less than symbols

  • Arrows show the range of values that are allowed

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

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

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