Which Expression Is Equivalent To $\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}}$?A. $\frac{2x-8}{13-5x}$B. $\frac{-5x-8}{2x-8}$C. $\frac{-5x-4}{x-2}$D. $\frac{13-5x}{2x-8}$

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Introduction

In algebra, complex fractions can be a challenge to simplify. These fractions involve multiple levels of division and can be difficult to work with. In this article, we will explore how to simplify a complex fraction, using the expression 3x−2−52−4x−2\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}} as an example. We will also examine the different answer choices and determine which one is equivalent to the given expression.

Step 1: Simplify the Numerator

To simplify the given expression, we first need to focus on the numerator. The numerator is 3x−2−5\frac{3}{x-2}-5. To simplify this expression, we can start by finding a common denominator for the two terms. The common denominator is x−2x-2, so we can rewrite the expression as 3x−2−5(x−2)x−2\frac{3}{x-2}-\frac{5(x-2)}{x-2}.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the numerator
numerator = 3/(x-2) - 5

# Simplify the numerator
simplified_numerator = sp.simplify(numerator)
print(simplified_numerator)

Step 2: Simplify the Denominator

Next, we need to simplify the denominator. The denominator is 2−4x−22-\frac{4}{x-2}. To simplify this expression, we can start by finding a common denominator for the two terms. The common denominator is x−2x-2, so we can rewrite the expression as 2(x−2)x−2−4x−2\frac{2(x-2)}{x-2}-\frac{4}{x-2}.

# Define the denominator
denominator = 2 - 4/(x-2)

# Simplify the denominator
simplified_denominator = sp.simplify(denominator)
print(simplified_denominator)

Step 3: Combine the Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the final expression. The final expression is 3−5(x−2)x−22(x−2)−4x−2\frac{\frac{3-5(x-2)}{x-2}}{\frac{2(x-2)-4}{x-2}}.

# Define the final expression
final_expression = (3-5*(x-2))/(2*(x-2)-4)

# Simplify the final expression
simplified_final_expression = sp.simplify(final_expression)
print(simplified_final_expression)

Answer Choices

Now that we have simplified the given expression, we can examine the different answer choices and determine which one is equivalent to the given expression.

A. 2x−813−5x\frac{2x-8}{13-5x}

B. −5x−82x−8\frac{-5x-8}{2x-8}

C. −5x−4x−2\frac{-5x-4}{x-2}

D. 13−5x2x−8\frac{13-5x}{2x-8}

Conclusion

In conclusion, the given expression 3x−2−52−4x−2\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}} is equivalent to the simplified expression −5x−4x−2\frac{-5x-4}{x-2}. This is answer choice C.

Final Answer

Introduction

In our previous article, we explored how to simplify a complex fraction, using the expression 3x−2−52−4x−2\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}} as an example. We also examined the different answer choices and determined which one is equivalent to the given expression. In this article, we will provide a Q&A guide to help you better understand how to simplify complex fractions in algebra.

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: Why is it difficult to simplify complex fractions?

A: Complex fractions can be difficult to simplify because they involve multiple levels of division and can be difficult to work with.

Q: What are the steps to simplify a complex fraction?

A: To simplify a complex fraction, you need to follow these steps:

  1. Simplify the numerator and denominator separately.
  2. Combine the numerator and denominator to get the final expression.
  3. Simplify the final expression.

Q: How do I simplify the numerator and denominator separately?

A: To simplify the numerator and denominator separately, you need to find a common denominator for the two terms. The common denominator is the least common multiple (LCM) of the two denominators.

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.

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

A: To find the LCM of two numbers, you can use the following steps:

  1. List the multiples of each number.
  2. Identify the smallest multiple that both numbers have in common.

Q: What are some common mistakes to avoid when simplifying complex fractions?

A: Some common mistakes to avoid when simplifying complex fractions include:

  • Not finding a common denominator for the numerator and denominator.
  • Not simplifying the numerator and denominator separately.
  • Not combining the numerator and denominator to get the final expression.

Q: How do I know if I have simplified a complex fraction correctly?

A: To know if you have simplified a complex fraction correctly, you need to check your work by plugging in different values for the variable.

Q: What are some real-world applications of simplifying complex fractions?

A: Simplifying complex fractions has many real-world applications, including:

  • Calculating probabilities and statistics.
  • Solving systems of equations.
  • Modeling real-world phenomena.

Conclusion

In conclusion, simplifying complex fractions in algebra can be a challenging task, but with the right steps and techniques, you can simplify even the most complex fractions. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in simplifying complex fractions and apply your skills to real-world problems.

Final Answer

The final answer is that simplifying complex fractions in algebra is a valuable skill that can be applied to many real-world problems. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying complex fractions and achieve your goals.

Additional Resources

For more information on simplifying complex fractions, check out the following resources:

  • Khan Academy: Simplifying Complex Fractions
  • Mathway: Simplifying Complex Fractions
  • Wolfram Alpha: Simplifying Complex Fractions

FAQs

Q: What is the difference between a complex fraction and a simple fraction?

A: A simple fraction is a fraction that contains only one fraction in its numerator or denominator. A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: How do I simplify a complex fraction with multiple levels of division?

A: To simplify a complex fraction with multiple levels of division, you need to follow the steps outlined in this article and simplify the numerator and denominator separately.

Q: Can I use a calculator to simplify complex fractions?

A: Yes, you can use a calculator to simplify complex fractions, but it's always a good idea to check your work by plugging in different values for the variable.