Solving Rational Inequalities

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  • 0:04 Rational Inequalities
  • 1:32 Steps for Solving
  • 4:35 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Rational inequalities often show up in areas such as engineering, medicine, and finance. This lesson will use a real-life example to demonstrate the step-by-step process used to solve these types of inequalities.

Rational Inequalities

Suppose you've just started building bicycles that you are going to sell. Your projected profit can be represented by the following function:


In this function, x is the number of bicycles you sell. You obviously want to make money, so you want to know how many units you'll have to sell so that your profit will be greater than zero. In other words, you want to solve the following inequality:


This is a rational inequality. A rational inequality is an inequality that contains a rational expression, where a rational expression is a ratio of two polynomials. That is, a rational expression is of the form R(x) / Q(x), where R(x) and Q(x) are polynomials and Q(x) is not zero. The general form of a rational inequality has a rational expression on the left-hand side of the inequality and a 0 on the right-hand side of the inequality.


To solve the inequality and see how many bicycles you must sell to make a profit, you may be tempted to multiply both sides of the inequality by x to get rid of the denominator, but resist that temptation!

When dealing with inequalities, if we multiply both sides by a negative number, we must switch the direction of the inequality symbol. Since the denominator contains a variable, we don't know if it will be positive or negative, so we don't know if we need to switch the direction of the symbol. Thus, in rational inequalities, we can't eliminate the denominators by multiplying both sides by an expression containing a variable.

So how do we solve rational inequalities?

Steps for Solving

To solve a rational inequality, we follow these steps:

  1. Put the inequality in general form.
  2. Set the numerator and denominator equal to zero and solve. The values you get are called critical values. The critical values of a function are where the function is undefined or equal to 0. When the numerator is 0, the function is 0. When the denominator is 0, the function is undefined.
  3. Plot the critical values on a number line, breaking the number line into intervals.
  4. Take a test number from each interval and plug it into the original inequality. If it makes a true statement, then the interval from which it came is in the solution. If it makes a false statement, then the interval from which it came is not in the solution.
  5. Determine if the endpoints of the intervals in the solution should be included in the intervals.

Let's put an end to the suspense of how many bicycles you must sell to make some money by taking the rational inequality representing your profit using these steps.

The first thing we want to do is put it in general form. Good news! In this instance, this has been done for us. We have a rational expression on the left-hand side of the inequality and a 0 on the right-hand side.


Next, we want to set the numerator and denominator equal to 0 and solve.


This gives us the critical values x = -30, x = 30, and x = 0. Let's plot these on a number line, breaking the number line into intervals.


Now let's find the test values from each interval. We can use any number that falls in each interval as a test value. It's usually a good idea to use numbers that are easy to work with. In this case, let's use -40 from interval I, -1 from interval II, 1 from interval III, and 40 from interval IV.


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