1 The inequality symbols
Recall the definitions of the inequality symbols in Key Point 11:
So for example,
A number line is often a helpful way of picturing inequalities. Given two numbers and , if then will be to the right of on the number line as shown in Figure 9.
Figure 9 :
Note from Figure 10 that , and .
Figure 10
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 or . In general if then . If then will be to the left of on the number line.
Example 33
Rewrite the inequality using only the ‘greater than’ sign, .
Solution
can be written as
Example 34
Rewrite the inequality using only the ‘less than’ sign, .
Solution
can be written as .
Sometimes two inequalities are combined into a single statement. Consider for example the statement . This is a compact way of writing ‘ and ’. Now is equivalent to and so means is greater than 3 but less than 6.
Inequalities obey simple rules when used in conjunction with arithmetical operations:
Key Point 12
- Adding or subtracting the same quantity from both sides of an inequality leaves the inequality symbol unchanged.
- Multiplying or dividing both sides by a positive number leaves the inequality unchanged.
- Multiplying or dividing both sides by a negative number reverses the inequality.
For example, since , by adding to both sides we can state
for any value of . For example (with ) . Further, by multiplying both sides of by we can state provided is positive. However, if 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 , multiplying both sides by gives .
Task!
Find the result of multiplying both sides of the inequality by .
The modulus or magnitude sign is sometimes used with inequalities. For example represents the set of all numbers whose actual size, irrespective of sign, is less than 1. This means any value between and 1. Thus
Similarly means all numbers whose size, irrespective of sign, is greater than 4. This means any value greater than 4 or less than . Thus
In general, if is a positive number:
Exercises
-
State which of the following statements are true and which are false.
- ,
- ,
- ,
- ,
- ,
- ,
In questions 2-9 rewrite each of the statements without using a modulus sign:
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- .
-
- F
- F
- T
- F
- T
- T
- T
- or
- or
- or , in fact any .