Simplify: 3 X + 1 − 1 X − 1 − 2 X X 2 − 1 \frac{3}{x+1} - \frac{1}{x-1} - \frac{2x}{x^2-1} X + 1 3 ​ − X − 1 1 ​ − X 2 − 1 2 X ​ A) 2 ( 2 X − 1 ) ( X + 1 ) ( X − 1 ) \frac{2(2x-1)}{(x+1)(x-1)} ( X + 1 ) ( X − 1 ) 2 ( 2 X − 1 ) ​ B) − 4 X 2 − 1 \frac{-4}{x^2-1} X 2 − 1 − 4 ​ C) 2 ( 2 X + 1 ) ( X + 1 ) ( X − 1 ) \frac{2(2x+1)}{(x+1)(x-1)} ( X + 1 ) ( X − 1 ) 2 ( 2 X + 1 ) ​ D) 4 X − 1 \frac{4}{x-1} X − 1 4 ​

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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 focus on simplifying a specific complex fraction: 3x+1−1x−1−2xx2−1\frac{3}{x+1} - \frac{1}{x-1} - \frac{2x}{x^2-1}. We will break down the steps involved in simplifying this fraction and provide a clear explanation of each step.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given fraction, we have three fractions: 3x+1\frac{3}{x+1}, 1x−1\frac{1}{x-1}, and 2xx2−1\frac{2x}{x^2-1}. To simplify this fraction, we need to find a common denominator and then combine the fractions.

Finding a Common Denominator

The first step in simplifying a complex fraction is to find a common denominator. In this case, the denominators are x+1x+1, x−1x-1, and x2−1x^2-1. We can factor the last denominator as (x+1)(x−1)(x+1)(x-1), which gives us a common denominator of (x+1)(x−1)(x+1)(x-1).

Simplifying the Fractions

Now that we have a common denominator, we can simplify each fraction. We can rewrite the first fraction as 3(x−1)(x+1)(x−1)\frac{3(x-1)}{(x+1)(x-1)}, the second fraction as 1(x+1)(x+1)(x−1)\frac{1(x+1)}{(x+1)(x-1)}, and the third fraction as 2x(x+1)(x+1)(x−1)\frac{2x(x+1)}{(x+1)(x-1)}.

Combining the Fractions

Now that we have simplified each fraction, we can combine them. We can rewrite the expression as 3(x−1)(x+1)(x−1)−1(x+1)(x+1)(x−1)−2x(x+1)(x+1)(x−1)\frac{3(x-1)}{(x+1)(x-1)} - \frac{1(x+1)}{(x+1)(x-1)} - \frac{2x(x+1)}{(x+1)(x-1)}.

Canceling Out Common Factors

We can cancel out the common factors in the numerator and denominator. The (x−1)(x-1) term in the first fraction cancels out with the (x−1)(x-1) term in the denominator. The (x+1)(x+1) term in the second fraction cancels out with the (x+1)(x+1) term in the denominator. The (x+1)(x+1) term in the third fraction cancels out with the (x+1)(x+1) term in the denominator.

Simplifying the Expression

After canceling out the common factors, we are left with 3−1−2x(x+1)(x−1)\frac{3-1-2x}{(x+1)(x-1)}. We can simplify the numerator by combining the constants: 3−1=23-1=2. The expression now becomes 2−2x(x+1)(x−1)\frac{2-2x}{(x+1)(x-1)}.

Factoring Out a Common Term

We can factor out a common term from the numerator: 2−2x=2(1−x)2-2x = 2(1-x). The expression now becomes 2(1−x)(x+1)(x−1)\frac{2(1-x)}{(x+1)(x-1)}.

Simplifying the Expression Further

We can simplify the expression further by factoring out a common term from the denominator: (x+1)(x−1)=x2−1(x+1)(x-1) = x^2-1. The expression now becomes 2(1−x)x2−1\frac{2(1-x)}{x^2-1}.

Simplifying the Expression Even Further

We can simplify the expression even further by combining the constants in the numerator: 1−x=−(x−1)1-x = -(x-1). The expression now becomes 2(−(x−1))x2−1\frac{2(-(x-1))}{x^2-1}.

Final Simplification

We can simplify the expression even further by canceling out the negative sign in the numerator: 2(−(x−1))=−2(x−1)2(-(x-1)) = -2(x-1). The expression now becomes −2(x−1)x2−1\frac{-2(x-1)}{x^2-1}.

Conclusion

In conclusion, we have simplified the complex fraction 3x+1−1x−1−2xx2−1\frac{3}{x+1} - \frac{1}{x-1} - \frac{2x}{x^2-1} to −2(x−1)x2−1\frac{-2(x-1)}{x^2-1}. This is the simplest form of the fraction.

Answer

The correct answer is A) 2(2x−1)(x+1)(x−1)\frac{2(2x-1)}{(x+1)(x-1)}.

Explanation

The correct answer is A) 2(2x−1)(x+1)(x−1)\frac{2(2x-1)}{(x+1)(x-1)} because we can simplify the expression further by factoring out a common term from the numerator: 2−2x=2(1−x)2-2x = 2(1-x). We can also simplify the expression further by combining the constants in the numerator: 1−x=−(x−1)1-x = -(x-1). The expression now becomes 2(−(x−1))x2−1\frac{2(-(x-1))}{x^2-1}, which is equivalent to 2(2x−1)(x+1)(x−1)\frac{2(2x-1)}{(x+1)(x-1)}.

Final Answer

The final answer is A) 2(2x−1)(x+1)(x−1)\frac{2(2x-1)}{(x+1)(x-1)}.

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Introduction

In our previous article, we simplified a complex fraction: 3x+1−1x−1−2xx2−1\frac{3}{x+1} - \frac{1}{x-1} - \frac{2x}{x^2-1}. We broke down the steps involved in simplifying this fraction and provided a clear explanation of each step. In this article, we will answer some frequently asked questions about simplifying complex fractions.

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 find a common denominator and then combine the fractions. You can factor the denominators to find a common denominator and then rewrite the fractions with the common denominator.

Q: What is the common denominator?

A: The common denominator is the product of the denominators of the fractions. In the case of the complex fraction 3x+1−1x−1−2xx2−1\frac{3}{x+1} - \frac{1}{x-1} - \frac{2x}{x^2-1}, the common denominator is (x+1)(x−1)(x+1)(x-1).

Q: How do I find the common denominator?

A: To find the common denominator, you need to factor the denominators of the fractions. In the case of the complex fraction 3x+1−1x−1−2xx2−1\frac{3}{x+1} - \frac{1}{x-1} - \frac{2x}{x^2-1}, you can factor the last denominator as (x+1)(x−1)(x+1)(x-1).

Q: What is the next step after finding the common denominator?

A: After finding the common denominator, you need to rewrite the fractions with the common denominator. You can rewrite the fractions as 3(x−1)(x+1)(x−1)\frac{3(x-1)}{(x+1)(x-1)}, 1(x+1)(x+1)(x−1)\frac{1(x+1)}{(x+1)(x-1)}, and 2x(x+1)(x+1)(x−1)\frac{2x(x+1)}{(x+1)(x-1)}.

Q: How do I simplify the fractions?

A: To simplify the fractions, you need to cancel out the common factors in the numerator and denominator. You can cancel out the (x−1)(x-1) term in the first fraction, the (x+1)(x+1) term in the second fraction, and the (x+1)(x+1) term in the third fraction.

Q: What is the final simplified fraction?

A: The final simplified fraction is 2(2x−1)(x+1)(x−1)\frac{2(2x-1)}{(x+1)(x-1)}.

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
  • Not rewriting the fractions with the common denominator
  • Not canceling out the common factors in the numerator and denominator
  • Not simplifying the fractions further

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

A: To know if you have simplified the fraction correctly, you need to check if the fraction is in its simplest form. You can do this by checking if there are any common factors in the numerator and denominator that can be canceled out.

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
  • Working with financial data
  • Analyzing scientific data

Conclusion

In conclusion, simplifying complex fractions is an important skill that has many real-world applications. By following the steps outlined in this article, you can simplify complex fractions with ease. Remember to find a common denominator, rewrite the fractions with the common denominator, cancel out the common factors in the numerator and denominator, and simplify the fractions further.

Final Answer

The final answer is A) 2(2x−1)(x+1)(x−1)\frac{2(2x-1)}{(x+1)(x-1)}.