Completely Simplify The Rational Expression $6-\frac{(x-1)}{(x+1)}$.A. $\frac{5x+7}{x+1}$ B. $\frac{12x}{x+1}$ C. $\frac{7x+7}{x}$ D. $\frac{7x+7}{x+1}$

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Introduction

Simplifying rational expressions is a crucial skill in algebra, and it requires a deep understanding of the underlying concepts. In this article, we will focus on simplifying the rational expression $6-\frac{(x-1)}{(x+1)}$. We will break down the problem step by step, and by the end of this article, you will be able to simplify this expression with ease.

Understanding Rational Expressions

Before we dive into the problem, let's quickly review what rational expressions are. A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by canceling out common factors in the numerator and denominator.

Step 1: Simplify the Expression Inside the Parentheses

The first step in simplifying the rational expression $6-\frac{(x-1)}{(x+1)}$ is to simplify the expression inside the parentheses. We can do this by combining the terms inside the parentheses.

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

Step 2: Find a Common Denominator

To simplify the expression further, we need to find a common denominator for the two fractions. In this case, the common denominator is $x+1$.

$$\frac{x}{x+1} - \frac{1}{x+1} = \frac{x-1}{x+1}$$

Step 3: Rewrite the Original Expression

Now that we have simplified the expression inside the parentheses, we can rewrite the original expression.

$$6-\frac{(x-1)}{(x+1)} = 6 - \frac{x-1}{x+1}$$

Step 4: Simplify the Expression

To simplify the expression further, we can rewrite it as a single fraction.

$$6 - \frac{x-1}{x+1} = \frac{6(x+1) - (x-1)}{x+1}$$

Step 5: Simplify the Numerator

Now that we have rewritten the expression as a single fraction, we can simplify the numerator.

$$6(x+1) - (x-1) = 6x + 6 - x + 1 = 5x + 7$$

Step 6: Write the Final Answer

Now that we have simplified the numerator, we can write the final answer.

$$\frac{6(x+1) - (x-1)}{x+1} = \frac{5x+7}{x+1}$$

The final answer is $\boxed{\frac{5x+7}{x+1}}$.

Conclusion

In this article, we simplified the rational expression $6-\frac{(x-1)}{(x+1)}$ step by step. We started by simplifying the expression inside the parentheses, then found a common denominator, and finally simplified the expression to get the final answer. By following these steps, you can simplify any rational expression with ease.

Discussion

Which of the following is the correct answer?

A. $\frac{5x+7}{x+1}$ B. $\frac{12x}{x+1}$ C. $\frac{7x+7}{x}$ D. $\frac{7x+7}{x+1}$

The correct answer is A. $\frac{5x+7}{x+1}$.

Final Answer

The final answer is $\boxed{\frac{5x+7}{x+1}}$.

Introduction

Simplifying rational expressions is a crucial skill in algebra, and it requires a deep understanding of the underlying concepts. In this article, we will answer some frequently asked questions (FAQs) about simplifying rational expressions. We will cover a range of topics, from basic concepts to advanced techniques.

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by canceling out common factors in the numerator and denominator.

Q: How do I simplify a rational expression?

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

1. Simplify the expression inside the parentheses.
2. Find a common denominator for the fractions.
3. Rewrite the expression as a single fraction.
4. Simplify the numerator.
5. Write the final answer.

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a number that can be expressed as a fraction, such as 3/4 or 22/7. A rational expression, on the other hand, is a fraction that contains variables and/or constants in the numerator and/or denominator.

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

A: Yes, you can simplify a rational expression with a variable in the denominator. However, you need to be careful not to divide by zero.

Q: How do I handle a rational expression with a negative exponent?

A: To handle a rational expression with a negative exponent, you need to rewrite the expression with a positive exponent. For example, if you have the expression $x^{-2}$, you can rewrite it as $\frac{1}{x^2}$.

Q: Can I simplify a rational expression with a fraction in the numerator?

A: Yes, you can simplify a rational expression with a fraction in the numerator. However, you need to find a common denominator for the fractions and then simplify the expression.

Q: How do I know when to simplify a rational expression?

A: You should simplify a rational expression whenever possible. Simplifying rational expressions can help you:

• Reduce the complexity of the expression
• Make it easier to solve equations and inequalities
• Improve the accuracy of your calculations

Q: Can I use a calculator to simplify a rational expression?

A: Yes, you can use a calculator to simplify a rational expression. However, you need to be careful not to make mistakes when entering the expression into the calculator.

Q: How do I check my work when simplifying a rational expression?

A: To check your work when simplifying a rational expression, you can:

• Plug in values for the variables to see if the expression simplifies correctly
• Use a calculator to check the expression
• Compare your answer to the original expression to see if it is equivalent

Conclusion

In this article, we answered some frequently asked questions (FAQs) about simplifying rational expressions. We covered a range of topics, from basic concepts to advanced techniques. By following these tips and techniques, you can simplify rational expressions with ease.

Discussion

Do you have any questions about simplifying rational expressions? Ask us in the comments below!

Final Answer

The final answer is $\boxed{\frac{5x+7}{x+1}}$.