# 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

Logo Source

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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