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Mar 9, 2025 by ADMIN 614 views

**Solving Rational Equations with Variables in the Denominator**

Introduction

Rational equations are a fundamental concept in algebra, and solving them can be a challenging task, especially when the denominator contains a variable. In this article, we will focus on solving rational equations with variables in the denominator, using the given equation $\frac{8x}{x-1} = \frac{8}{x-1} + 4$ as an example.

Understanding Rational Equations

A rational equation is an equation that contains one or more rational expressions, which are fractions that contain variables in the numerator or denominator. Rational expressions can be added, subtracted, multiplied, or divided, but they cannot be simplified in the same way as polynomial expressions.

Step 1: Identify the Variable in the Denominator

The first step in solving a rational equation with a variable in the denominator is to identify the variable. In the given equation, the variable is $x$ , which appears in the denominator of both fractions.

Step 2: Multiply Both Sides by the Least Common Multiple (LCM)

To eliminate the variable in the denominator, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is $(x-1)^2$ .

\frac{8x}{x-1} = \frac{8}{x-1} + 4
\\
8x(x-1) = 8(x-1) + 4(x-1)^2
\\
8x(x-1) = 8(x-1) + 4(x^2-2x+1)
\\
8x(x-1) = 8(x-1) + 4x^2-8x+4
\\
8x^2-8x = 8(x-1) + 4x^2-8x+4
\\
8x^2-8x = 8x-8 + 4x^2-8x+4
\\
8x^2-8x = 4x^2-8x+4
\\
8x^2-4x^2 = 4
\\
4x^2 = 4
\\
x^2 = 1
\\
x = \pm 1

Step 3: Check for Extraneous Solutions

After solving the equation, we need to check for extraneous solutions. An extraneous solution is a solution that is not valid in the original equation. In this case, we need to check if $x = 1$ is an extraneous solution.

\frac{8x}{x-1} = \frac{8}{x-1} + 4
\\
x = 1
\\
\frac{8(1)}{1-1} = \frac{8}{1-1} + 4
\\
\frac{8}{0} = \frac{8}{0} + 4
\\
\text{This is undefined}

Since $x = 1$ is an extraneous solution, we need to exclude it from the solution set.

Conclusion

Solving rational equations with variables in the denominator requires careful attention to detail and a thorough understanding of algebraic concepts. By following the steps outlined in this article, we can solve rational equations with variables in the denominator and identify any extraneous solutions.

Final Answer

The solution set is $\boxed{\{-1\}}$ .

Discussion

Rational equations with variables in the denominator can be challenging to solve, but with practice and patience, you can master this skill. Remember to always check for extraneous solutions and to be careful when multiplying both sides of the equation by the LCM.

Additional Resources

For more information on solving rational equations with variables in the denominator, check out the following resources:

Khan Academy: Solving Rational Equations
Mathway: Solving Rational Equations
Wolfram Alpha: Solving Rational Equations

FAQs

Q: What is a rational equation? A: A rational equation is an equation that contains one or more rational expressions.

Q: How do I solve a rational equation with a variable in the denominator? A: To solve a rational equation with a variable in the denominator, multiply both sides of the equation by the LCM of the denominators.

Q: What is an extraneous solution? A: An extraneous solution is a solution that is not valid in the original equation.

Q: What is a rational equation?

A: A rational equation is an equation that contains one or more rational expressions. Rational expressions are fractions that contain variables in the numerator or denominator.

Q: How do I identify a rational equation?

A: To identify a rational equation, look for fractions or rational expressions in the equation. If the equation contains a fraction with a variable in the numerator or denominator, it is a rational equation.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. In the context of rational equations, the LCM is the smallest multiple that the denominators of the fractions have in common.

Q: How do I find the LCM of two or more numbers?

A: To find the LCM of two or more numbers, list the multiples of each number and find the smallest multiple that they all have in common.

Q: What is the difference between a rational expression and a rational equation?

A: A rational expression is a fraction that contains variables in the numerator or denominator. A rational equation is an equation that contains one or more rational expressions.

Q: How do I solve a rational equation with a variable in the denominator?

A: To solve a rational equation with a variable in the denominator, multiply both sides of the equation by the LCM of the denominators.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that is not valid in the original equation. Extraneous solutions can occur when the solution is substituted into the original equation and the result is undefined.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, substitute the solution into the original equation and check if it is defined. If the solution is substituted into the original equation and the result is undefined, it is an extraneous solution.

Q: What is the importance of checking for extraneous solutions?

A: Checking for extraneous solutions is important because it ensures that the solution is valid and not a result of a mistake in the solution process.

Q: Can I use a calculator to solve rational equations with variables in the denominator?

A: Yes, you can use a calculator to solve rational equations with variables in the denominator. However, be careful when using a calculator to check for extraneous solutions, as some calculators may not be able to handle undefined results.

Q: Are there any special cases to consider when solving rational equations with variables in the denominator?

A: Yes, there are special cases to consider when solving rational equations with variables in the denominator. For example, if the denominator is zero, the equation is undefined and cannot be solved.

Q: Can I use algebraic methods to solve rational equations with variables in the denominator?

A: Yes, you can use algebraic methods to solve rational equations with variables in the denominator. However, be careful when using algebraic methods, as they may not always be the most efficient or effective way to solve the equation.

Q: Are there any online resources available to help me learn how to solve rational equations with variables in the denominator?

A: Yes, there are many online resources available to help you learn how to solve rational equations with variables in the denominator. Some popular resources include Khan Academy, Mathway, and Wolfram Alpha.

Conclusion

Solving rational equations with variables in the denominator can be a challenging task, but with practice and patience, you can master this skill. Remember to always check for extraneous solutions and to be careful when multiplying both sides of the equation by the LCM. If you have any further questions or need additional help, don't hesitate to ask.