# 5 Inequalities: greater than or less than?

There have been three occasions in these last two weeks when checks have been made to see whether a result is greater than or less than some other value. These were when:

- calculating the BMI and determining whether the person was overweight or underweight
- determining from footprints whether a person was walking or running
- checking whether a phone had been used for more than 300 minutes.

In all these situations the statements could have also been written mathematically using inequality symbols.

This checking of whether values are greater than, or less than, some limit happens frequently – occasions when you may see this could be safety issues or age limits. For example, medicines may have to be stored at a temperature of 25 °C or less; child train tickets can be bought for children who are over 5 but under 16 years old.

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

**>**greater than**≥**greater than or equal to**<**less than**≤**less than or equal to

If you have difficulty remembering these symbols, you can think of them as arrows that point to the smaller number or note that ‘<’ looks like an ‘L’, which stands for ‘less than’.

The symbols are read from left to right. For example, ‘11 > 9’ is read as ‘11 is greater than 9’; the cost of a holiday in pounds < 1000 is read as ‘the cost of a holiday 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 would be written as ‘10 > 5’. On the number line, because –4 lies to the left of –3, –4 is less than –3; this would be written as ‘–4 < –3’.

Similarly, the instructions for the medicine that has to be stored at a temperature of 25 °C or less could be written as ‘Medicine storage temperature in °C ≤ 25’.

Sometimes, it can be helpful to draw a number line to visualise this kind of information. For example, the ages that children are eligible for the child train fare are from their fifth birthday up to, but not including, their sixteenth birthday. This means that the age has to be greater than or equal to 5 and less than 16. This range is shown on the number line below. Does this help you to visualise the information?

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 is then written as ‘5 ≤ age for child train fare < 16’.

Note carefully the format in which this last inequality is written. The variable that is being described – that is, the age for a child train fare – is always in the centre when defining a range of values with an upper and lower limit. This is the maths convention that is followed with ranges of values. This can be a little confusing, as you have to turn your logic around for the lower limit. You are in effect saying that 5 is less than, or equal to, a child train fare, rather than a child train fare is greater than, or equal to, 5. This amounts to exactly the same thing, but how you may usually think about the lower limit is turned on its head!

Now put these ideas into practice in this final activity for the week.

## Activity _unit5.5.1 Activity 10 Inequalities

- a.Which symbol (< or >) should go in the blank spaces below? Click on ‘reveal comment’ if you would like a hint to get going.
- i.4__7
- ii.18__10
- iii.3__–2

### Discussion

Write the inequality using words first and then convert this to the correct symbol.

### Answer

a.

- i.4 < 7
- ii.18 > 10
- iii.3 > –2

- b.Work out what the following mathematical statements mean. Write your answers as full sentences and be as precise as you can.
- i.Balance in account > 0.
- ii.Speed (in mph) on motorway ≤ 70.
- iii.18 ≤ age (in years) ≤ 50.

### Answer

- b.Your answers may be worded slightly differently, but they should still have the same meaning:
- i.The balance in the account is greater than zero.
- ii.The speed on the motorway is less than or equal to 70 mph.
- iii.The age in years is between 18 and 50, inclusively.

- c.Rewrite the following sentences as statements using the inequality symbols:
- i.The refrigerator 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.

### Answer

- c.Your answers may be worded slightly differently again, but they should still have the same meaning:
- i.Refrigerator temperature (in ºC) < 4.
- ii.Number of people on committee ≥ 5.
- iii.20 < BMI < 25

As well as in the examples given here, inequalities are also an important part of many computer programs. So you can see that maths pops up in all sorts of areas of study and everyday lives.

This is the end of this section on inequalities and the week as a whole. You may have found some ideas that were new to you here and others that were more familiar. Whatever the case for you, hopefully you feel even more confident with your maths skills now.