# Inequalities

By M. Bourne

This chapter explains the properties of inequalities and then goes on to show how to solve linear and non-linear inequalities. Finally, we see how to solve inequalities that involve absolute values.

## Why study inequalities?

Logo of Devon & Exeter

Racial Equality Council, England

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All men are born equal,

but some are more equal

than others...

(The root of all
prejudice)

Inequalities are very common in daily life. For example:

**Thermostats**in cars cause a valve to open when the engine gets hot (say more than `95°"C"`), allowing water to circulate and cool the engine down. We can express this condition using an inequality: `T > 95°"C"`. If the engine is getting too cool (say `T < 85°"C"`), the thermostat closes again, reducing the water circulation.- A
**voltage regulator**in a TV will typically accept a voltage range from 110V to 240V. We could write the range for the voltage*V*as `110 ≤ V ≤ 240`.

**Obesity**is usually defined in terms of the Body Mass Index (BMI).- `"BMI" < 18.5` is underweight
- `18.5 < "BMI" < 24.9` is normal weight
- `25.0 < "BMI" < 29.9` is overweight
- `30.0 < "BMI" < 39.9` is obese
- `"BMI" > 40.0` is severely (or morbidly) obese

[The BMI is the mass of the person in kg divided by the square of the person's height in m.]

## In this Chapter

- 1. Properties of Inequalities - the difference between "`<`", "`>`", "`≤`" , etc.
- 2. Solving Linear Inequalities - problems like: `3 - 2x ≥ 15`
- 3. Solving Non-Linear Inequalties - more advanced examples
- 4. Inequalities Involving Absolute Values - with examples

Let's first learn some of the Properties of Inequalities ».

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