After watching this video lesson, you will be able to solve both equations and inequalities that involve rational functions. Learn the steps involved in solving them as well as what to watch out for.

## Rational Functions

Hello there. And welcome to this video lesson on **rational functions**. Remember, that our rational functions are those functions that are the division of two polynomials. We can have simple rational functions like 3/*x* or more complicated ones like (*x* – 2) / (*x*^2 – 4*x* + 4).

In this video lesson, we are going to learn how to solve equations and inequalities that involve these rational functions. Yes, you will see an equation or inequality that has rational functions on one or both sides of the equation or inequality. You will learn the steps you need to take to solve these problems, and then I will give you some pointers on what to keep in mind. You’ll come across these types of problems more and more often as you progress in your math, so knowing how to solve these types of equations will serve you well. And hey, you’ll be able to help out your friends who need help with these problems.

## Solving a Rational Equation

Let’s begin by looking at solving an equation with rational functions in it. I’ll give you a simple equation for the sake of keeping the steps clear. The steps are the same, no matter how big the rational functions are. What you need to know to get started is your algebra and how to solve various types of polynomials, the most important being how to factor quadratics.

Imagine that you are taking a test, and you’re given this problem:

You need to solve it for *x*. How can you go about doing so? You think back to algebra and you remember that when you see an equal sign between two fractions like this, to solve it, you need to cross multiply. So, you multiply the numerator of one side with the denominator of the other side and then vice versa. Now you have this:

To continue solving, you now need to bring all your terms to one side. So you subtract the 9 from both sides:

The reason you do this is because you have an exponent connected with your variable. If you solved it without moving all your terms over to one side, you might miss some solutions. Now that all your terms are on one side, you can now go ahead and solve like you did in algebra. You see this is a quadratic, so you go ahead and factor to solve:

Your answers are 3 and -3. Because we are dealing with rational functions here, we need to plug these solutions back into our original problem just to check that they work. Plugging 3 back into the original problem, we get 3 / 3 = 3 / 3. That turns into 1 = 1, which checks out. Plugging -3 back into the original problem, we get 3 / -3 = -3 / 3. That turns into -1 = -1, which checks out as well. Both check out, so we have two solutions: 3 and -3.

## Solving a Rational Inequality

Now let’s look at solving an inequality involving rational functions. Imagine that you are still taking the test, and now you see this problem:

To solve this problem, you now need to analyze the rational function. To do this, you need to find where the rational function equals 0 and where the rational functions is undefined. The rational function equals 0 when the numerator equals 0, and it is undefined when the denominator equals 0. So, you split your rational function into its numerator and denominator parts and solve each for 0.

Solving the numerator for 0, you get *x* = 0. Setting the denominator equal to 0, you get *x* – 2 = 0. Solving this for *x*, you get *x* = 2.

These numbers that you just got split our function into intervals. By plugging in test numbers within each interval, we can find out in which interval (or intervals) our rational function is less than or equal to 0.

Our first interval is from negative to 0. We can plug in -1 as a test number. We get -1 / (-1 – 2) = -1 / -3 = 1/3. This is a positive number, so this interval is not a solution.

Our next interval is from 0 to 2. We can plug in 1.5 or 3/2. We get (3/2) / (3/2 – 2) = 3 / (2(-1/2)) = 3 / (-1) = -3. Aha! This interval is negative and fits my inequality, so this interval is a solution. What about the last interval? Let’s check.

Our last interval is from 2 to positive infinity. We can plug in 3 to test. We get 3 / (3 – 2) = 3/1 = 3. Nope, this interval is not a solution because we got a positive number greater than 0.

Our solution then is the interval from 0 to 2 inclusive. We can write this as [0, 2], where the square brackets show that we are including the numbers 0 and 2. We would use parentheses if we weren’t including those numbers; for example, if our inequality was less than instead of less than or equal to.

## Points to Keep in Mind

We’ve now seen how to solve both equations and inequalities that involve rational functions. Here are some pointers to keep in mind.

- Write out each step. Don’t try to do it in your head. The math can get tricky with bigger rational functions. You want to be able to see all your steps nicely on a paper so that if you make a mistake on a multiplication or sign, you will be able to see it and correct it easily.
- Test all your solutions. It is possible that some of your solutions are false solutions. It is always a good idea to plug your solutions into the original function to make sure that they’re correct. Also, if you made a mistake somewhere along the way, this check will tell you.

## Lesson Summary

Let’s review what we’ve learned now.

**Rational functions** are those functions that are the division of two polynomials. To solve an equation involving rational functions, we cross multiply the numerators and denominators. Then we move all our terms to one side. Then we use our algebra skills to solve.

To solve an inequality involving rational functions, we set our numerator and denominator to 0 and solve them separately. This will give us numbers that divide our function into intervals. We then take a test number within each interval to find out which interval meets our inequality. The intervals that meet our inequality are the solution intervals.

## Learning Outcomes

After you have finished with this lesson, you should be able to:

- Recognize rational functions
- Solve rational equations and inequalities
- Implement tips for working with these types of problems