1 The inequality symbols

Recall the definitions of the inequality symbols in Key Point 11:

Key Point 11

The symbols > , < , ≥ , ≤ are called inequalities

>  means: ‘is greater than’ , ≥  means: ‘is greater than or equal to’

<  means: ‘is less than’ , ≤  means: ‘is less than or equal to’

So for example,

8 > 7 9 ≥ 2 − 2 < 3 7 ≤ 7

A number line is often a helpful way of picturing inequalities. Given two numbers a and b , if b > a then b will be to the right of a on the number line as shown in Figure 9.

Figure 9 :

A number line segment with labeled points \(a\) and \(b\), indicating that \(b\) is to the right of \(a\).

Long description

Note from Figure 10 that − 3 > − 5 , 4 > − 2 and 8 > 5 .

Figure 10

A number line marked with integers from negative five to eight.

Long description

Inequalities can always be written in two ways. For example in English we can state that 8 is greater than 7, or equivalently, that 7 is less than 8. Mathematically we write 8 > 7 or 7 < 8 . In general if b > a then a < b . If a < b then a will be to the left of b on the number line.

Example 33

Rewrite the inequality − 2 5 < x using only the ‘greater than’ sign, > .

Solution

− 2 5 < x can be written as x > − 2 5

Example 34

Rewrite the inequality 5 > x using only the ‘less than’ sign, < .

Solution

5 > x can be written as x < 5 .

Sometimes two inequalities are combined into a single statement. Consider for example the statement 3 < x < 6 . This is a compact way of writing ‘ 3 < x and x < 6 ’. Now 3 < x is equivalent to x > 3 and so 3 < x < 6 means x is greater than 3 but less than 6.

Inequalities obey simple rules when used in conjunction with arithmetical operations:

Key Point 12
  1. Adding or subtracting the same quantity from both sides of an inequality leaves the inequality symbol unchanged.
  2. Multiplying or dividing both sides by a positive number leaves the inequality unchanged.
  3. Multiplying or dividing both sides by a negative number reverses the inequality.

For example, since 8 > 5 , by adding k to both sides we can state

8 + k > 5 + k

for any value of k . For example (with k = − 3 ) 8 − 3 > 5 − 3 . Further, by multiplying both sides of 8 > 5 by k we can state 8 k > 5 k   provided k is positive. However, 8 k < 5 k   if k is negative.

We emphasise that the inequality sign is reversed when multiplying both sides by a negative number. A common mistake is to forget to reverse the inequality symbol. For example if 8 > 5 , multiplying both sides by − 1 gives − 8 < − 5 .

Task!

Find the result of multiplying both sides of the inequality − 18 < 9 by − 3 .

54 > − 27

The modulus or magnitude sign is sometimes used with inequalities. For example x < 1 represents the set of all numbers whose actual size, irrespective of sign, is less than 1. This means any value between − 1 and 1. Thus

x < 1  means  − 1 < x < 1

Similarly x > 4 means all numbers whose size, irrespective of sign, is greater than 4. This means any value greater than 4 or less than − 4 . Thus

x > 4  means  x > 4  or  x < − 4

In general, if k is a positive number:

Key Point 13

x < k means − k < x < k

x > k means x > k or x < − k

Exercises
  1. State which of the following statements are true and which are false.
    1. 4 > 9 ,
    2. 4 > 4 ,
    3. 4 ≥ 4 ,
    4. 0.001 < 1 0 − 5 ,
    5. − 19 < 100 ,
    6. − 19 > − 20 ,
    7. 0.001 ≤ 1 0 − 3

    In questions 2-9 rewrite each of the statements without using a modulus sign:

  2.    | x | < 2 ,
  3.    | x | < 5 ,
  4.    | x | ≤ 7.5 ,
  5.    x − 3 < 2 ,
  6.    | x − a | < 1 ,
  7.    | x | > 2 ,
  8.    | x | > 7.5 ,
  9.    | x | ≥ 0 .
    1. F
    2. F
    3. T
    4. F
    5.   T
    6. T
    7. T
  2. − 2 < x < 2
  3. − 5 < x < 5
  4. − 7.5 ≤ x ≤ 7.5
  5. − 2 < x − 3 < 2
  6. − 1 < x − a < 1
  7. x > 2 or x < − 2
  8. x > 7.5 or x < − 7.5
  9. x ≥ 0 or x ≤ 0 , in fact any x .