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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?

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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

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

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