Solving Linear Inequalities (AQA GCSE Maths): Revision Note

Solving linear inequalities

What is an 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 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

    • = 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 an 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

How do I represent linear inequalities using set notation?

• We use curly brackets and a colon in set notation. means "x is in the set ..."

• For example; if x is greater than 3, then in set notation,

• However, if x is between two values, then the two end values must be written in separate sets, using the intersection symbol, 

  • For example, if x is greater than 3 and less than or equal to 5, then in set notation,

• Similarly, if x is less than one value or greater than another (disjoint), then the two end values must be written in separate sets using the union symbol, 

  • For example, if x is less than 3 or greater than or equal to 5, then in set notation,