Fractional inequalities

Fractional Inequalities: An Overview

  • Fractional inequalities involve inequalities where the variable x appears in the denominator.

  • Understanding how to handle fractional inequalities is crucial as they feature in various areas of mathematics.

  • They are a bit more complex than linear or quadratic inequalities, but they can be solved by using similar techniques.

Basic Principle

  • As with standard inequalities, the aim is to isolate the variable, x, on one side of the inequality and keep the number on another side.

  • One fundamental rule to remember is: when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality changes.

Steps to Solving Fractional Inequalities

  • Firstly, rearrange the inequality so that it is in the form f(x) < 0 or f(x) > 0.

  • Once the inequality is in this form, the next step is to find any values of x for which f(x) = 0. These values are the critical values.

  • Plotting these critical values on a number line separates it into intervals.

  • Pick a test point in each interval. Substituting this back into f(x) and checking the sign will provide information about the sense of the inequality in that interval.

  • If the sign of the test point satisfies the inequality, then all points in that interval are part of the solution set.

The Zero Factor Property

  • A helpful shortcut called the zero factor property states that if a product of factors equals zero, then at least one of those factors must be zero.

  • Using this property can significantly streamline solving fractional inequalities, as we no longer have to factorise and solve each term individually.

Caution Points

  • Remember to consider the undefined values where the denominator is zero. Exclude these from your solution set.

  • For the inequality to hold, the sense (i.e., the direction of the inequality) can change depending on whether you’re dealing with a “less than” or a “greater than” inequality.

Remember, as with any mathematical concept, practice is essential. Work through examples and keep your steps logical and orderly. Utilising these techniques will help you master the art of solving fractional inequalities.