Polynomial Inequalities: Definition & Examples

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Polynomial inequalities show up in real world applications when intervals of numbers are involved. In this lesson, you'll learn what polynomial inequalities are and how to work with them.

Polynomial Inequalities

Suppose you're trying to catch a cab in the city. You have no more than $20 to spend, and the cabs charge a flat rate of $2.00 plus $0.70 per mile. If x represents the number of miles you ride in the cab, then the cost of the cab ride would be 2 + 0.7x. That is, the flat rate of $2.00 plus $0.70 times the number of miles you ride in the cab. In this situation, you want the cab ride to cost less than the $20 you have to spend. You can represent this mathematically as follows:

2 + 0.7x < 20

This is a polynomial inequality. Polynomial inequalities are inequalities expressed with a polynomial on one side of the inequality symbol and zero on the other side. Some examples of polynomial inequalities are shown below:

Examples of Polynomial Inequalities
polynomial inequalities 1

These types of inequalities can be used to answer questions about real-world situations, such as your city cab ride. Suppose you want to know how many miles you can travel without exceeding your spending limit. To find out, you solve the polynomial inequality for x to get x < 25.71. So, you can ride in the cab for about 25 miles without exceeding your spending limit.

Solutions to Polynomial Inequalities

The solution to the cab example is x < 25.71, which is an interval, or a set of numbers. The solution to a polynomial inequality consists of intervals that make the inequality true. In the example, x < 25.71 tells you that if you plug in any number that is less than 25.71 into the inequality, you'll get a true statement. For instance, the number 10 is less than 25.71, so it should make a true statement when plugged into your inequality.

1.) 2 + 0.7(10) < 20

2.) 2 + 7 < 20

3.) 9 < 20

Sure enough, this is a true statement because 9 is less than 20. On the other hand, if you plugged in a number greater than 25.71, you'd get a false statement, since the number doesn't fall in your interval solution. For instance, the number 30 is greater than 25.71, which would lead to a false statement, as shown below.

1.) 2 + 0.7(30) < 20

2.) 2 + 21 < 20

3.) 23 < 20

As expected, this is a false statement because 23 is not less than 20.

Solving Polynomial Inequalities

To solve polynomial inequalities, follow the steps below:

1.) Manipulate the inequality so you'll have a polynomial on one side of the inequality symbol and zero on the other side.

2.) Replace the inequality symbol with an equal symbol, and solve the equation.

3.) Plot the solution from step two on a number line.

4.) Take a test value from each of the intervals, making sure they're not equal to the endpoints of the intervals. Plug the test values into the original inequality.

5.) If the test value leads to a true statement, then the interval from which it came is the solution. If the test value leads to a false statement, then the interval from which it came is not the solution.

Now, try applying these steps to your city cab ride.

1.) To get zero on one side of the inequality, subtract 20 from both sides:

2 + 0.7x < 20

0.7x - 18 < 0

2.) Find all values of x that make 0.7x - 18 = 0:

0.7x - 18 = 0

0.7x = 18

x = 18 / 0.7 = 25.71

3.) Plot x = 25.71 on the number line. In the image below, you'll see that it's broken into two intervals: x < 25.71 and x > 25.71.

Number Line Broken into Intervals
polynomial inequality 3

4.) Choose a test number from each of the intervals, such as 10 and 30. The number 10 comes from the interval x < 25.71, and the number 30 comes from the interval x > 25.71.

5.) The number 10 makes for a true statement when plugged into the inequality, while 30 makes for a false statement. Therefore, the solution is the interval from which 10 originated: x < 25.71.

Practice Problem

Let's try one more example. Suppose you own a business and want to determine the price of a product that would make your revenue greater than your cost using the following inequality:

45 - 9p < 5p - p^2

1.) Get zero on one side:

45 - 9p < 5p - p^2

p^2 - 9p - 5p + 45 <0

p^2 - 14p + 45 < 0

2.) Find values of p that make p^2 - 14p + 45 = 0:

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