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?

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.]

racial equality logo
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)

In this Chapter

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