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.]

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 ».