Back To Course

Math 101: College Algebra12 chapters | 95 lessons | 11 flashcard sets

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 70,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Kathryn Maloney*

Kathryn teaches college math. She holds a master's degree in Learning and Technology.

A rational equation is one that contains fractions. Yes, we will be finding a common denominator that has 'x's. But no worries! Together we will use a process that will help us solve rational equations every time!

A rational equation is an equation that contains fractions with *x*s in the numerator, denominator or both. Here is an example of a rational equation: (4 / (*x* + 1)) - (3 / (*x* - 1)) = -2 / (*x*^2 - 1).

Let's think back for a moment about solving an equation with a fraction. 1/3 *x* = 8. We think of the 3 in the denominator as being a prisoner, and we want to release it. To set the 3 free, we multiply both sides of the equation by 3. Think of it as 3 letting both sides of the equation know he's leaving. 3 (1/3 *x*) = 8 (3).

This process freed our denominator and got rid of the fraction - *x* = 24. It is also the process we use to solve rational equations with one extra step. In rational equations, sometimes our solution may look good, but they carry a virus; that is, they won't work in our equation. These are called extraneous solutions. The steps to solve a rational equation are:

- Find the common denominator.
- Multiply everything by the common denominator.
- Simplify.
- Check the answer(s) to make sure there isn't an extraneous solution.

Let's solve a couple together.

Example number one: solve. Remember to check for extraneous solutions. (3 / (*x* + 3)) + (4 / (*x* - 2)) = 2 / (*x* + 3).

Our first step is to figure out the terms that need to be released from the denominators. I look at 3 / (*x* + 3). I write down (*x* + 3) as one of my common denominators. I look at 4 / (*x* - 2). I write down (*x* - 2) as another part of my common denominator. I look at 2 / (*x* + 3). Since I already have (*x* + 3) written in my denominator, I don't need to duplicate it.

Next, we multiply everything by our common denominator - (*x*+3)(*x*-2). This is how that will look: ((3(*x* + 3)(*x* - 2)) / (*x* + 3)) + ((4(*x* + 3)(*x* - 2)) / (*x* - 2)) = (2(*x* + 3)(*x* - 2)) / (*x* + 3))

It isn't easy for the denominators to be released; there is a battle, and like terms in the numerator and denominator get canceled (or slashed). Slash (or cancel) all of the (*x* + 3)s and (*x* - 2)s in the denominator and numerator. Our new equation looks like: 3(*x* - 2) + 4(*x* + 3) = 2(*x* - 2).

Distribute to simplify: (3*x* - 6) + (4*x* + 12) = 2*x* - 4. Collect like terms and solve. 3*x* + 4*x* = 7*x*, -6 + 12 = 6. We end up with 7*x* + 6 = 2*x* - 4.

Subtract 2*x* from both sides: 7*x* - 2*x* = 5*x*. Subtracting from the other side just cancels out the 2*x*, and we get 5*x* + 6 = -4. Subtract 6 from both sides: -4 - 6 = -10. Again, subtracting 6 will cancel out the +6, so we end up with 5*x* = - 10. Divide by 5 on both sides, and we cancel out the 5 and give us *x* = - 2. It turns out *x* = - 2.

The reason we check our answers is that sometimes we get a virus, or, in math terms, extraneous solutions. To check, I replace all the *x*s with -2: (3 / (-2 + 3)) + (4 / (-2 - 2)) = (2 / (-2 + 3)). Let's simplify: (3 / 1) + (4 / -4) = (2 / 1). Since 3 + -1 = 2 is true, *x* = - 2 is the solution!

Example number two: solve. Remember to check for extraneous solutions. (4 / (*x* + 1)) - (3 / (*x* - 1)) = -2 / (*x*^2 - 1).

First we need to release our denominators. To release our denominators, we write down every denominator we see. I have found the easiest way to do this is to first factor, if needed, then list the factors. *x*^2 - 1 = (*x* + 1)(*x* - 1).

Our new equation looks like this: (4 / (*x* + 1)) - (3 / (*x* - 1)) = -2 / (*x* + 1)(*x* - 1).

I look at 4 / (*x* + 1). I write down (*x* + 1) as one of my common denominators. I look at 3 / (*x* - 1). I write down (*x* - 1) as another part of my common denominator. I look at -2 / (*x* + 1)(*x* - 1). Since I already have those written in my denominator, I don't need to duplicate them. So my common denominator turns out to be (*x* + 1)(*x* - 1).

Kathryn, why aren't we using the factors of *x*^2 - 1? Great question! We already have (*x* + 1) and (*x* - 1) being released. We don't need to do it twice.

Now we multiply each part of the equation by the common denominator - (*x* + 1)(*x* - 1). Think of this as the key to the prison: (4 (*x* + 1)(*x* -1) / (*x* + 1)) - (3 (*x* + 1) (*x* - 1) / (*x* - 1)) = -2 (*x* + 1)(*x* - 1) / (*x* + 1)(*x* - 1).

It isn't easy for the denominators to be released; there is a battle, and like terms get canceled (or slashed)! Slash (or cancel) all of the (*x* + 1)s and (*x* - 1)s in the denominator and numerator. This leaves us with 4(*x* - 1) - 3 (*x* + 1) = -2.

Now we need to solve for *x*. Distribute 4 into (*x* - 1) and -3 into (*x* + 1). (4*x* - 4) - (3*x* - 3) = -2. Collect like terms: *x* - 7 = - 2. Add 7 to both sides of the equal sign: *x* = 5.

It looks like our answer is 5, but we need to double-check. I replace all the *x*s with 5 and simplify. It turns out 5 works, and it is the solution to our equation. And so our solution checks!

The steps to solving a rational equation are:

- Find the common denominator.
- Multiply everything by the common denominator.
- Simplify.
- Check the answer(s) to make sure there isn't an extraneous solution.

To unlock this lesson you must be a Study.com Member.

Create your account

Already a member? Log In

BackDid you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
5 in chapter 8 of the course:

Back To Course

Math 101: College Algebra12 chapters | 95 lessons | 11 flashcard sets

- MTLE Pedagogy - Secondary (Grades 5-12): Study Guide & Practice
- MTLE Pedagogy - Elementary (Grades K-6): Study Guide & Practice
- Computer Science 113: Programming in Python
- DSST Information Guide
- GACE Early Childhood Special Education General Curriculum: Practice & Study Guide
- Required Assignments for Communications 301
- Required Assignments for Psychology 306
- Computing with Numbers
- Introduction to Python Programming
- Object-Oriented Programming
- How to Prepare for the CTEL
- CTEL Test Score Information
- CTEL Test Accommodations
- CTEL Test Retake Policy
- CSET Test Day Preparation
- How to Study for the VCLA Test
- Can You Use a Calculator on the CBEST?

- The New Right: Definition & Movement
- The Role of ELL Teachers as a Resource & Advocate
- Civil War Border States: Definition & Significance
- Overview of Literacy Development Research
- Third-Party Tools for Social Selling: Types & Examples
- ANSI C: History, Formation & Structure
- Overriding Derived Classes in C++ Programming
- Using Typography in Visual Storytelling
- Quiz & Worksheet - Indian Ethnic Groups
- Quiz & Worksheet - Analyzing Assessment Results in ELL
- Quiz & Worksheet - Legal Protections for ELL Students
- Quiz & Worksheet - Differentiation for Teaching ELL
- Quiz & Worksheet - The Natural Learning Approach in ESL
- Flashcards - Introduction to Research Methods in Psychology
- Flashcards - Clinical Assessment in Psychology

- Business Law in Sales
- Middle School World History: Homeschool Curriculum
- Prentice Hall Geometry: Online Textbook Help
- 9th Grade English: High School
- 10th Grade English: Credit Recovery
- MTEL Middle School Math/Science: Functions
- Saxon Algebra 2: Perimeter and Circumference
- Quiz & Worksheet - HR Department's Role
- Quiz & Worksheet - Hedge Fund Characteristics & Structure
- Quiz & Worksheet - Surface Tension
- Quiz & Worksheet - What Are Economies of Scale?
- Quiz & Worksheet - Creating Energy from Various Sources

- What is JavaScript? - Function & Uses
- Indentured Servants in Jamestown: Definition & Overview
- Oregon Science Standards
- New York State Science Standards for Grade 4
- ELL Services in Illinois
- Homeschool vs. Public School Statistics
- Kansas Science Standards for 1st Grade
- Homeschooling in Nevada
- What are the NBPTS Standards?
- PMP Certification Exam Pass Rate
- 504 Plans in Arizona
- Transportation Lesson Plan

Browse by subject