# 7 Inequalities

In this course, there have been three occasions when checks have been made to see whether the result is greater than or less than some other value. The first case was in calculating the BMI and determining whether the person was overweight or underweight; the second case was in determining whether a person was walking or running from his footprints; and the third case was in checking whether a phone had been used for more than 30 minutes. Checking whether values are greater than or less than some limit happens frequently in everyday problems, particularly in safety limits but elsewhere too. For example, medicines may have to be stored at a temperature of 25 °C or less. Child rail tickets can be bought for children who are 5 or more years old but less than 16 years old.

Rather than writing out ‘greater than’ or ‘less than’, some shorthand notation is often used as shown below.

## Inequalities

> greater than

≥ greater than or equal to

< less than

≤ less than or equal to

If you have difficulty interpreting these symbols, you can think of them as arrows which point to the smaller number.

The symbols can be read from left to right. For example, 11 > 9 is read as ‘11 is greater than 9’ and holiday cost (in £) < 1000 is read as ‘holiday cost is less than £1000’.

To use the symbols in your own writing, decide what you want to say first, then use the symbol. For example, since 10 is greater than 5, this could be written as 10 > 5 or since on the number line ^{−}4 lies to the left of ^{−}3, ^{−}4 is less than ^{−}3 and this could be written as

^{−}4 < ^{−}3.

Similarly, the instructions for the medicine could be written as:

medicine temperature in °C ≤ 25

Sometimes, you may find it helpful to draw a number line to visualise this sort of information. For example, the ages which children can claim the child rail fare are from their fifth birthday up to but not including their sixteenth birthday. This means that their age has to be greater than or equal to 5 and less than 16.

Note how drawing a diagram here helps you to see what is happening.

The empty circle means that this number (16) is not included and the filled-in circle means that this number (5) is included in the interval. This can be written as:

5 ≤ age for child rail fare < 16.

## Activity 24: Inequalities

a.Which symbol (< or >) should appear in each of the boxes below.

i.4 □ 7

ii.18 □ 10

iii.3 □

^{−}2

b.Explain what the following mathematical statements mean.

i.Balance in account > 0.

ii.Speed (in mph) on motorway ≤ 70.

iii.18 ≤ age (in years) ≤ 50.

c.Express the following conditions using the inequality symbols.

i.The fridge temperature should be less than 4 °C.

ii.There must be at least five people on the committee.

iii.For an ideal weight, a person's BMI should be greater than 20 and less than 25.

### Discussion

a.

i.4 < 7

ii.18 > 10

iii.3 >

^{−}2

b.

i.The balance in the account is greater than zero, or in credit.

ii.The speed on the motorway is less than or equal to 70 mph.

iii.The age is between 18 and 50 years inclusive.

c.

i.Fridge temperature (in ºC) < 4.

ii.Number of people on committee ≥ 5.

iii.20 < BMI < 25.

Inequalities are also used a lot in computer programs to check whether conditions have been fulfilled. For example, if the balance in your bank account is negative, you may be prevented from withdrawing money from a cash machine.