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

Complex fractions can be intimidating, but with the right approach, they can be simplified with ease. In this article, we will explore the process of simplifying complex fractions, using the expression 3x−2−52−4x−2\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}} as an example. We will break down the expression into manageable parts, simplify each part, and then combine them to arrive at the final answer.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given expression, we have a fraction in the numerator and a fraction in the denominator. To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Simplifying the Numerator

The numerator of the given expression is 3x−2−5\frac{3}{x-2}-5. To simplify this expression, we need to find a common denominator for the two terms. The common denominator is x−2x-2, so we can rewrite the expression as:

3x−2−5=3x−2−5(x−2)x−2\frac{3}{x-2}-5 = \frac{3}{x-2}-\frac{5(x-2)}{x-2}

Now, we can combine the two fractions by adding their numerators:

3x−2−5(x−2)x−2=3−5(x−2)x−2\frac{3}{x-2}-\frac{5(x-2)}{x-2} = \frac{3-5(x-2)}{x-2}

Expanding the numerator, we get:

3−5(x−2)x−2=3−5x+10x−2\frac{3-5(x-2)}{x-2} = \frac{3-5x+10}{x-2}

Simplifying the numerator further, we get:

3−5x+10x−2=−5x+13x−2\frac{3-5x+10}{x-2} = \frac{-5x+13}{x-2}

Simplifying the Denominator

The denominator of the given expression is 2−4x−22-\frac{4}{x-2}. To simplify this expression, we need to find a common denominator for the two terms. The common denominator is x−2x-2, so we can rewrite the expression as:

2−4x−2=2(x−2)x−2−4x−22-\frac{4}{x-2} = \frac{2(x-2)}{x-2}-\frac{4}{x-2}

Now, we can combine the two fractions by subtracting their numerators:

2(x−2)x−2−4x−2=2(x−2)−4x−2\frac{2(x-2)}{x-2}-\frac{4}{x-2} = \frac{2(x-2)-4}{x-2}

Expanding the numerator, we get:

2(x−2)−4x−2=2x−4−4x−2\frac{2(x-2)-4}{x-2} = \frac{2x-4-4}{x-2}

Simplifying the numerator further, we get:

2x−4−4x−2=2x−8x−2\frac{2x-4-4}{x-2} = \frac{2x-8}{x-2}

Combining the Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to arrive at the final expression:

−5x+13x−22x−8x−2\frac{\frac{-5x+13}{x-2}}{\frac{2x-8}{x-2}}

To divide fractions, we need to invert the denominator and multiply:

−5x+13x−22x−8x−2=−5x+13x−2⋅x−22x−8\frac{\frac{-5x+13}{x-2}}{\frac{2x-8}{x-2}} = \frac{-5x+13}{x-2} \cdot \frac{x-2}{2x-8}

Now, we can cancel out the common factors:

−5x+13x−2⋅x−22x−8=−5x+132x−8\frac{-5x+13}{x-2} \cdot \frac{x-2}{2x-8} = \frac{-5x+13}{2x-8}

Simplifying the Final Expression

The final expression is −5x+132x−8\frac{-5x+13}{2x-8}. We can simplify this expression by factoring out the greatest common factor (GCF) of the numerator and denominator. The GCF of −5x+13-5x+13 and 2x−82x-8 is −1-1, so we can rewrite the expression as:

−5x+132x−8=−(5x−13)2x−8\frac{-5x+13}{2x-8} = \frac{-(5x-13)}{2x-8}

Now, we can factor out the GCF of the numerator:

−(5x−13)2x−8=−(5x−13)2(x−4)\frac{-(5x-13)}{2x-8} = \frac{-(5x-13)}{2(x-4)}

Conclusion

In this article, we simplified the complex fraction 3x−2−52−4x−2\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}} by breaking it down into manageable parts, simplifying each part, and then combining them to arrive at the final answer. We used the order of operations (PEMDAS) to simplify the numerator and denominator, and then combined them to arrive at the final expression. The final expression is 13−5x2x−8\frac{13-5x}{2x-8}.

Final Answer

The final answer is 13−5x2x−8\boxed{\frac{13-5x}{2x-8}}.

Comparison with Answer Choices

Let's compare the final answer with the answer choices:

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}

The final answer matches answer choice D.

Conclusion

In conclusion, we simplified the complex fraction 3x−2−52−4x−2\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}} by breaking it down into manageable parts, simplifying each part, and then combining them to arrive at the final answer. We used the order of operations (PEMDAS) to simplify the numerator and denominator, and then combined them to arrive at the final expression. The final expression is 13−5x2x−8\frac{13-5x}{2x-8}, which matches answer choice D.

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Introduction

In our previous article, we explored the process of simplifying complex fractions, using the expression 3x−2−52−4x−2\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}} as an example. We broke down the expression into manageable parts, simplified each part, and then combined them to arrive at the final answer. In this article, we will answer some common questions related to simplifying complex fractions.

Q&A

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: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: A simple fraction is a fraction that does not contain any fractions in its numerator or denominator. A complex fraction, on the other hand, contains one or more fractions in its numerator or denominator.

Q: Can I simplify a complex fraction by canceling out common factors?

A: Yes, you can simplify a complex fraction by canceling out common factors. However, you need to be careful not to cancel out any factors that are not common to both the numerator and denominator.

Q: How do I know when to use the order of operations to simplify a complex fraction?

A: You should use the order of operations to simplify a complex fraction whenever you encounter a fraction that contains one or more fractions in its numerator or denominator.

Q: Can I simplify a complex fraction by multiplying or dividing both the numerator and denominator by the same value?

A: Yes, you can simplify a complex fraction by multiplying or dividing both the numerator and denominator by the same value. However, you need to be careful not to change the value of the fraction.

Q: What is the final answer to the expression 3x−2−52−4x−2\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}}?

A: The final answer to the expression 3x−2−52−4x−2\frac{\frac{3}{x-2}-5}{2-\frac{4}{x-2}} is 13−5x2x−8\frac{13-5x}{2x-8}.

Common Mistakes to Avoid

When simplifying complex fractions, there are several common mistakes to avoid:

  • Not following the order of operations (PEMDAS)
  • Canceling out common factors that are not common to both the numerator and denominator
  • Multiplying or dividing both the numerator and denominator by the same value without changing the value of the fraction
  • Not simplifying the numerator and denominator separately before combining them

Conclusion

In conclusion, simplifying complex fractions can be a challenging task, but with the right approach, it can be done with ease. By following the order of operations (PEMDAS) and simplifying the numerator and denominator separately, you can arrive at the final answer. Remember to avoid common mistakes such as not following the order of operations, canceling out common factors that are not common to both the numerator and denominator, and multiplying or dividing both the numerator and denominator by the same value without changing the value of the fraction.

Final Tips

  • Practice simplifying complex fractions regularly to build your skills and confidence.
  • Use the order of operations (PEMDAS) to simplify complex fractions.
  • Simplify the numerator and denominator separately before combining them.
  • Avoid common mistakes such as not following the order of operations, canceling out common factors that are not common to both the numerator and denominator, and multiplying or dividing both the numerator and denominator by the same value without changing the value of the fraction.

Conclusion

In conclusion, simplifying complex fractions is a valuable skill that can be applied to a wide range of mathematical problems. By following the order of operations (PEMDAS) and simplifying the numerator and denominator separately, you can arrive at the final answer. Remember to practice regularly and avoid common mistakes to become proficient in simplifying complex fractions.