Interpreting Inequalities (Cambridge International IGCSE Mathematics)

Interpreting 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 (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

Worked Example List all the integer values of $x$ that satisfy

$-4\leq x<2$

Answer:

Integer values are whole numbers -4 ≤ x shows that x includes -4, so this is the first integer

x = -4*

x < 2 shows that x does not include 2 Therefore the last integer is x = 1

x = 1*

For the answer, list all the integers from -4 to 1 Remember integers can be zero and negative

$x=-4,-3,-2,-1,0,1$