AQA A Level Maths: Pure复习笔记2.4.1 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. 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