Add The Following Expressions:$\[ \frac{-x-1}{x^2-1} + \frac{x+3}{4x+4} \\]Enter Your Answer In Simplest Form.

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 given expression as an example. We will break down the steps involved in simplifying the expression and provide a clear, step-by-step guide on how to do it.

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 two fractions: $\frac{-x-1}{x^2-1}$ and $\frac{x+3}{4x+4}$. To simplify this expression, we need to find a common denominator and then combine the two fractions.

Step 1: Factor the Denominators

The first step in simplifying the expression is to factor the denominators. The denominator of the first fraction is $x^2-1$, which can be factored as $(x-1)(x+1)$. The denominator of the second fraction is $4x+4$, which can be factored as $4(x+1)$.

# Factored Denominators
## First Fraction
### Denominator: (x-1)(x+1)
## Second Fraction
### Denominator: 4(x+1)

Step 2: Find a Common Denominator

To combine the two fractions, we need to find a common denominator. The common denominator is the product of the two denominators: $(x-1)(x+1)(4)(x+1)$. This can be simplified to $4(x-1)(x+1)^2$.

# Common Denominator
## Common Denominator: 4(x-1)(x+1)^2

Step 3: Rewrite the Fractions

Now that we have a common denominator, we can rewrite the fractions with the common denominator.

# Rewritten Fractions
## First Fraction
### Numerator: (-x-1)(4)(x+1)
### Denominator: 4(x-1)(x+1)^2
## Second Fraction
### Numerator: (x+3)(x-1)
### Denominator: 4(x-1)(x+1)^2

Step 4: Combine the Fractions

Now that we have rewritten the fractions with the common denominator, we can combine them.

# Combined Fractions
## Numerator: (-x-1)(4)(x+1) + (x+3)(x-1)
## Denominator: 4(x-1)(x+1)^2

Step 5: Simplify the Numerator

The final step is to simplify the numerator. We can do this by combining like terms.

# Simplified Numerator
## Numerator: -4x^2 - 4x - 4x - 4 + 4x^2 - x - 3
## Denominator: 4(x-1)(x+1)^2

Step 6: Final Simplification

After combining like terms, we can simplify the expression further.

# Final Simplification
## Numerator: -7x - 7
## Denominator: 4(x-1)(x+1)^2

Conclusion

Simplifying complex fractions requires a step-by-step approach. By factoring the denominators, finding a common denominator, rewriting the fractions, combining the fractions, simplifying the numerator, and finally simplifying the expression, we can simplify complex fractions with ease. In this article, we used the given expression as an example and provided a clear, step-by-step guide on how to simplify it.

Final Answer

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Introduction

In our previous article, we explored the process of simplifying complex fractions, using the given expression as an example. In this article, we will answer some of the most frequently asked questions about simplifying complex fractions.

Q: What is a complex fraction?

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

Q: Why do we need to simplify complex fractions?

Simplifying complex fractions is important because it makes the expression easier to work with and understand. It also helps to eliminate any unnecessary complexity, making it easier to solve equations and inequalities.

Q: How do I simplify a complex fraction?

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

1. Factor the denominators.
2. Find a common denominator.
3. Rewrite the fractions with the common denominator.
4. Combine the fractions.
5. Simplify the numerator.
6. Final simplification.

Q: What is the common denominator?

The common denominator is the product of the two denominators. It is the denominator that both fractions have in common.

Q: How do I find the common denominator?

To find the common denominator, you need to multiply the two denominators together.

Q: What if the denominators are not factorable?

If the denominators are not factorable, you can use the least common multiple (LCM) to find the common denominator.

Q: Can I simplify a complex fraction with a variable in the denominator?

Yes, you can simplify a complex fraction with a variable in the denominator. However, you need to be careful when simplifying the expression, as the variable may affect the final result.

Q: What if I have a complex fraction with multiple variables in the denominator?

If you have a complex fraction with multiple variables in the denominator, you can simplify the expression by factoring the denominators and finding a common denominator.

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

Yes, you can use a calculator to simplify complex fractions. However, it's always a good idea to check your work by hand to ensure that the calculator is giving you the correct answer.

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

Some common mistakes to avoid when simplifying complex fractions include:

• Not factoring the denominators correctly
• Not finding the common denominator correctly
• Not rewriting the fractions with the common denominator correctly
• Not combining the fractions correctly
• Not simplifying the numerator correctly

Conclusion

Simplifying complex fractions can be a challenging task, but with the right approach and practice, it can become easier. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex fractions with ease.

Final Tips

• Always factor the denominators correctly
• Always find the common denominator correctly
• Always rewrite the fractions with the common denominator correctly
• Always combine the fractions correctly
• Always simplify the numerator correctly

By following these tips and practicing regularly, you can become proficient in simplifying complex fractions and tackle even the most challenging problems with confidence.