Simplify The Compound Fractional Expression:$\[ \frac{\frac{1}{x-1}+\frac{1}{x+3}}{x+1} \\]

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Introduction

Simplifying complex mathematical expressions is a crucial skill in mathematics, and it requires a deep understanding of algebraic manipulation and fraction simplification. In this article, we will focus on simplifying a compound fractional expression, which involves combining multiple fractions into a single expression. The given expression is ${ \frac{\frac{1}{x-1}+\frac{1}{x+3}}{x+1} }$. Our goal is to simplify this expression and present it in a more manageable form.

Understanding the Expression

Before we begin simplifying the expression, let's break it down and understand its components. The given expression consists of two fractions in the numerator and one fraction in the denominator. The numerator is a sum of two fractions, each with a different denominator. The denominator of the entire expression is a single fraction.

To simplify this expression, we need to find a common denominator for the fractions in the numerator and then combine them. After that, we can simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor (GCD).

Finding a Common Denominator

To find a common denominator for the fractions in the numerator, we need to identify the least common multiple (LCM) of their denominators. The denominators of the two fractions in the numerator are x−1x-1 and x+3x+3. The LCM of these two expressions is (x−1)(x+3)(x-1)(x+3).

Now that we have found the common denominator, we can rewrite the fractions in the numerator with this common denominator. We can do this by multiplying the numerator and denominator of each fraction by the necessary factors to obtain the common denominator.

Rewriting the Fractions

Let's rewrite the fractions in the numerator with the common denominator (x−1)(x+3)(x-1)(x+3).

{ \frac{1}{x-1} = \frac{(x+3)}{(x-1)(x+3)} \}

{ \frac{1}{x+3} = \frac{(x-1)}{(x-1)(x+3)} \}

Now that we have rewritten the fractions, we can combine them by adding their numerators.

Combining the Fractions

The numerator of the expression is now a single fraction with the common denominator (x−1)(x+3)(x-1)(x+3). We can combine the fractions by adding their numerators.

{ \frac{(x+3)}{(x-1)(x+3)} + \frac{(x-1)}{(x-1)(x+3)} = \frac{(x+3)+(x-1)}{(x-1)(x+3)} \}

Simplifying the numerator, we get:

{ \frac{(x+3)+(x-1)}{(x-1)(x+3)} = \frac{(2x+2)}{(x-1)(x+3)} \}

Simplifying the Expression

Now that we have combined the fractions in the numerator, we can simplify the entire expression by dividing the numerator and denominator by their GCD. The GCD of (2x+2)(2x+2) and (x−1)(x+3)(x-1)(x+3) is 2.

Dividing the numerator and denominator by 2, we get:

{ \frac{(2x+2)}{(x-1)(x+3)} = \frac{(x+1)}{(x-1)(x+3)/2} \}

However, we can simplify this expression further by canceling out the common factor of 2 in the denominator.

{ \frac{(x+1)}{(x-1)(x+3)/2} = \frac{(x+1)}{(x-1)(x+3)/2} \times \frac{2}{2} \}

Simplifying the expression, we get:

{ \frac{(x+1)}{(x-1)(x+3)/2} = \frac{(x+1) \times 2}{(x-1)(x+3)} \}

{ \frac{(x+1) \times 2}{(x-1)(x+3)} = \frac{(2x+2)}{(x-1)(x+3)} \}

However, we can simplify this expression further by canceling out the common factor of 2 in the numerator and denominator.

{ \frac{(2x+2)}{(x-1)(x+3)} = \frac{(x+1)}{(x-1)(x+3)/2} \}

Conclusion

In this article, we simplified a compound fractional expression by finding a common denominator, rewriting the fractions, combining them, and simplifying the resulting expression. The simplified expression is ${ \frac{(x+1)}{(x-1)(x+3)/2} }$. This expression is more manageable and easier to work with than the original compound fractional expression.

Final Answer

The final answer is (x+1)(x−1)(x+3)/2\boxed{\frac{(x+1)}{(x-1)(x+3)/2}}.

References

Note: The references provided are for general information and are not specific to the problem at hand. They are included to provide additional context and resources for readers who may be interested in learning more about algebraic manipulation and fraction simplification.

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Introduction

In our previous article, we simplified a compound fractional expression by finding a common denominator, rewriting the fractions, combining them, and simplifying the resulting expression. In this article, we will answer some frequently asked questions (FAQs) related to simplifying compound fractional expressions.

Q&A

Q: What is a compound fractional expression?

A: A compound fractional expression is a mathematical expression that consists of multiple fractions combined in a single expression. It can be a sum, difference, product, or quotient of fractions.

Q: How do I simplify a compound fractional expression?

A: To simplify a compound fractional expression, you need to follow these steps:

  1. Find a common denominator for the fractions in the expression.
  2. Rewrite the fractions with the common denominator.
  3. Combine the fractions by adding or subtracting their numerators.
  4. Simplify the resulting expression by dividing the numerator and denominator by their greatest common divisor (GCD).

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

A: The LCM of two expressions is the smallest expression that is a multiple of both expressions. To find the LCM of two expressions, you can list the multiples of each expression and find the smallest multiple that is common to both.

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

A: To find the LCM of two expressions, you can follow these steps:

  1. List the multiples of each expression.
  2. Find the smallest multiple that is common to both expressions.
  3. The LCM is the smallest multiple that is common to both expressions.

Q: What is the greatest common divisor (GCD) of two expressions?

A: The GCD of two expressions is the largest expression that divides both expressions without leaving a remainder. To find the GCD of two expressions, you can use the Euclidean algorithm.

Q: How do I find the GCD of two expressions?

A: To find the GCD of two expressions, you can follow these steps:

  1. Use the Euclidean algorithm to find the GCD of the two expressions.
  2. The GCD is the largest expression that divides both expressions without leaving a remainder.

Q: Can I simplify a compound fractional expression with a variable in the denominator?

A: Yes, you can simplify a compound fractional expression with a variable in the denominator. However, you need to be careful when simplifying the expression, as the variable may affect the GCD of the numerator and denominator.

Q: How do I simplify a compound fractional expression with a variable in the denominator?

A: To simplify a compound fractional expression with a variable in the denominator, you need to follow these steps:

  1. Find a common denominator for the fractions in the expression.
  2. Rewrite the fractions with the common denominator.
  3. Combine the fractions by adding or subtracting their numerators.
  4. Simplify the resulting expression by dividing the numerator and denominator by their GCD.

Conclusion

In this article, we answered some frequently asked questions related to simplifying compound fractional expressions. We covered topics such as finding a common denominator, rewriting fractions, combining fractions, and simplifying the resulting expression. We also discussed the importance of finding the least common multiple (LCM) and greatest common divisor (GCD) of two expressions.

Final Answer

The final answer is (x+1)(x−1)(x+3)/2\boxed{\frac{(x+1)}{(x-1)(x+3)/2}}.

References

Note: The references provided are for general information and are not specific to the problem at hand. They are included to provide additional context and resources for readers who may be interested in learning more about algebraic manipulation and fraction simplification.