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 < 90°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.]