Linear Inequations (SQA National 5 Maths): Revision Note

Solving linear inequations

What is an inequation?

• An inequation (also known as 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, ...

    • or any number greater than 5, like 5.25 or 97.8 or 5.0990195...

      • x doesn't need to be an integer

• Inequations 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,....

      • or 9.5 or -3

• When inequations cannot be equal, they are called strict inequations

  • > and < are strict inequations

    • x > 5 does not include 5 (strict)

    • x ⩾ 5 does include 5 (not strict)

How do I solve linear inequations?

• Solving linear inequations 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 inequation 

  • 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 variable terms onto one side

  • and all the number terms onto the other side

How do I find integers that satisfy inequations?

• Sometimes you may be interested in particular integers (whole numbers) that satisfy an inequation

• If you are given two end points then 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 you only wanted positive integers then just list x = 2, 1

• You can also find integers that satisfy two inequations

  • 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 

• Or you can find the smallest or largest integer

  • The smallest integer that satisfies x > 6.5 is 7