Inequalities

The aim of this document is to provide a short, self-assessment programme for students who wish to acquire a basic competence in the use of inequalities.

A number a is greater than a number b if a - b is positive. In symbols this is written as a > b.

Thus

2 > 1 because 2 - 1 = 1 is positive,

3 > - 1 because 3 - ( - 1) = 4 is positive,

BUT

- 1 > 2 is false because - 1 - 2 = - 3 is negative.

Example 1

Prove or disprove the following inequalities.

Solution

(a) As a decimal, 1/4 = 0.25 and so 0.4 - 1/4 = 0.4 - 0.25 = 0.15, which is positive. Thus 0.4 > 1/4 is true.

(b) Here (0.7) = 0.7 × 0.7 = 0.49. As a fraction 1/2 is 0.5. In this case, (0.7) - 1/2 = 0.49 - 0.5 = - 0.01, which is negative. This means that the inequality (0.7) > 1/2 is false.

For this latter example we would write (0.7) < 1/2, or in words, (0.7) is less than 1/2.

In general we say:

A number a is less than a number b if a - b is negative. In symbols this is written as a < b.

If a < b the b > a and vice versa.

Example 2

In each of the following pairs of numbers, use one of the symbols > or < to give the correct ordering of the numbers for the order in which they appear.

Solution

(a) Taking a = - 1 and b = 2 the difference a - b , becomes a - b = (-1) - 2 = - 3 , which is negative. The correct inequality is - 1 < 2.

(b) In decimal form 1/4 = 0.25 and 1/5 = 0.2. Since 0.25 - 0.2 = 0.05, and this is positive, the correct inequality is 1/4 > 1/5.

In addition to these two inequalities there are two further symbols, and . The first of these is read as greater than or equal to and the second as less than or equal to.

Exercise

For each of the following pairs of numbers use one of the symbols >, <, , to give the correct ordering for the order in which they appear.

Solutions

(a)

(b) (-1) = 1 and (-1/2) = 1/4 so (-1) > (-1/2).

(c) In decimal form 1/ 5 = 0.2 so 0.2 1/ 5 and 0.2 1/ 5 are both true.

(d) The solution to this can be obtained by converting the fractions to decimals as in previous cases. It may also be obtained using fractions, by writing both with the same denominator 6.

Then

which is negative. The correct inequality is therefore

Quiz

Determine which of the following inequalities is correct.

(a) 3 > 2, (b) 2 < 4, (c) 2 < 5, (d) 3 > 4

Solution

The solution to this is obtained from 3 = 9 and 2 = 8 and 9 > 8.