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. 5>3
  • <Less than e.g. -8<7
  • ≥ 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