Solving Linear Inequalities (Edexcel GCSE Maths: Foundation)

What is a linear inequality?

• An inequality tells you that something is greater than (>) or less than (<) something else
  • x > 5 means x is greater than 5 
    • x could be 6, 7, 8, 9, ...
• Inequalities may also include being equal (=) 
  • ⩾ means greater than or equal to
  • ⩽ means less than or equal to
    • x ⩽ 10 means x is less than or equal to 10
      • x could be 10, 9, 8, 7, 6, ...
• When they cannot be equal, they are called strict inequalities
  • > and < are strict inequalities
    • x > 5 does not include 5 (strict)
    • x ⩾ 5 does include 5 (not strict)

How do I find integers that satisfy inequalities?

• You may be given two end points and have to list the integer (whole number) values of x that satisfy the inequality
• Look at whether each end point is included or not 
  • 3 ⩽ x ⩽ 6
    • x = 3, 4, 5, 6
  • 3 ⩽ x < 6
    • x = 3, 4, 5
  • 3 < x ⩽ 6
    • x = 4, 5, 6
  • 3 < x < 6
    • x = 4, 5
• If only one end point is given, there are an infinite number of integers
  • x > 2
    • x = 3, 4, 5, 6, ...
  • x ⩽ 2
    • x = 2, 1, 0, -1, -2, ...
    • Remember zero and negative whole numbers are integers
    • If the question had said positive integers only then just list x = 2, 1
• You may be asked to find integers that satisfy two inequalities
  • 0 < x < 5 and x ⩾ 3
    • List separately: x = 1, 2, 3, 4 and x = 3, 4, 5, 6,  ...
    • Find the values that appear in both lists: x = 3, 4 
• If the question does not say x is an integer, do not assume x is an integer!
  • x > 3 actually means any value greater than 3
    • 3.1 is possible
    • π = 3.14159... is possible
• You may be asked to find the smallest or largest integer
  • The smallest integer that satisfies x > 6.5 is 7

How do I represent a linear inequality on a number line?

• The inequality -3 < x ≤ 4 is shown on a number line below

• Draw circles above the end points and connect them with a horizontal line
  • Leave an open circle for end points with strict inequalities, < or >
    • These end points are not included
  • Fill in a solid circle for end points with ≤ or ≥ inequalities
    • These end points are included

• Use a horizontal arrow for inequalities with one end point
  • x > 5 is an open circle at 5 with a horizontal arrow pointing to the right 

How do I solve a 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. 
    
• Never multiply or divide by a variable (x) as this could be positive or negative
• The safest way to rearrange is simply to add and subtract to move all the terms onto one side

How do I solve double inequalities?

• Inequalities such as can be solved by doing the same thing to all three parts of the inequality
  • Use the same rules as solving linear inequalities