Simplify The Expression: \[$\frac{x}{x+1} - 1\$\]Given: \[$x = \frac{1}{6}\$\]

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Introduction

In this article, we will simplify the given expression $\frac{x}{x+1} - 1$ using the provided value of $x = \frac{1}{6}$. We will use algebraic manipulation and substitution to simplify the expression and find its final value.

Understanding the Expression

The given expression is $\frac{x}{x+1} - 1$. This expression involves a fraction and a constant term. To simplify this expression, we need to find a common denominator for the fraction and the constant term.

Finding a Common Denominator

To find a common denominator, we can multiply the numerator and denominator of the fraction by the same value. In this case, we can multiply both the numerator and denominator of the fraction by $x+1$.

Simplifying the Expression

Using the common denominator, we can rewrite the expression as:

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

Combining the Fractions

Now that we have a common denominator, we can combine the two fractions into one:

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

Simplifying the Numerator

We can simplify the numerator by expanding the product and combining like terms:

$$x(x+1) - (x+1) = x^2 + x - x - 1 = x^2 - 1$$

Substituting the Value of $x$

Now that we have simplified the expression, we can substitute the value of $x = \frac{1}{6}$ into the expression:

$$\frac{x^2 - 1}{(x+1)^2} = \frac{\left(\frac{1}{6}\right)^2 - 1}{\left(\frac{1}{6} + 1\right)^2}$$

Evaluating the Expression

To evaluate the expression, we need to simplify the numerator and denominator separately:

$$\left(\frac{1}{6}\right)^2 - 1 = \frac{1}{36} - 1 = -\frac{35}{36}$$

$$\left(\frac{1}{6} + 1\right)^2 = \left(\frac{7}{6}\right)^2 = \frac{49}{36}$$

Final Answer

Now that we have simplified the numerator and denominator, we can substitute the values back into the expression:

$$\frac{-\frac{35}{36}}{\frac{49}{36}} = -\frac{35}{49}$$

Conclusion

In this article, we simplified the given expression $\frac{x}{x+1} - 1$ using the provided value of $x = \frac{1}{6}$. We used algebraic manipulation and substitution to simplify the expression and find its final value. The final answer is $-\frac{35}{49}$.

Frequently Asked Questions

• What is the given expression? The given expression is $\frac{x}{x+1} - 1$.
• What is the value of $x$? The value of $x$ is $\frac{1}{6}$.
• How do we simplify the expression? We simplify the expression by finding a common denominator, combining the fractions, simplifying the numerator, and substituting the value of $x$.
• What is the final answer? The final answer is $-\frac{35}{49}$.

Step-by-Step Solution

1. Find a common denominator for the fraction and the constant term.
2. Rewrite the expression using the common denominator.
3. Combine the two fractions into one.
4. Simplify the numerator by expanding the product and combining like terms.
5. Substitute the value of $x$ into the expression.
6. Simplify the numerator and denominator separately.
7. Substitute the values back into the expression.

Common Mistakes

• Not finding a common denominator for the fraction and the constant term.
• Not combining the fractions into one.
• Not simplifying the numerator correctly.
• Not substituting the value of $x$ into the expression.
• Not simplifying the numerator and denominator separately.

Real-World Applications

This problem has real-world applications in algebra and calculus. It can be used to model real-world situations, such as population growth and decay, and to solve problems involving fractions and algebraic expressions.

Further Reading

For further reading on this topic, we recommend the following resources:

• "Algebra and Trigonometry" by Michael Sullivan
• "Calculus" by Michael Spivak
• "Mathematics for the Nonmathematician" by Morris Kline

Conclusion

In conclusion, we simplified the given expression $\frac{x}{x+1} - 1$ using the provided value of $x = \frac{1}{6}$. We used algebraic manipulation and substitution to simplify the expression and find its final value. The final answer is $-\frac{35}{49}$.

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Frequently Asked Questions

Q: What is the given expression?

A: The given expression is $\frac{x}{x+1} - 1$.

Q: What is the value of $x$?

A: The value of $x$ is $\frac{1}{6}$.

Q: How do we simplify the expression?

A: We simplify the expression by finding a common denominator, combining the fractions, simplifying the numerator, and substituting the value of $x$.

Q: What is the final answer?

A: The final answer is $-\frac{35}{49}$.

Q: What is the common denominator for the fraction and the constant term?

A: The common denominator is $(x+1)^2$.

Q: How do we combine the two fractions into one?

A: We combine the two fractions into one by adding or subtracting their numerators while keeping the same denominator.

Q: How do we simplify the numerator?

A: We simplify the numerator by expanding the product and combining like terms.

Q: Why do we need to substitute the value of $x$ into the expression?

A: We need to substitute the value of $x$ into the expression to find the final answer.

Q: How do we simplify the numerator and denominator separately?

A: We simplify the numerator and denominator separately by performing the necessary operations.

Q: What are some common mistakes to avoid when simplifying the expression?

A: Some common mistakes to avoid when simplifying the expression include not finding a common denominator, not combining the fractions into one, not simplifying the numerator correctly, not substituting the value of $x$ into the expression, and not simplifying the numerator and denominator separately.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in algebra and calculus, such as modeling population growth and decay, and solving problems involving fractions and algebraic expressions.

Q: Where can I find further reading on this topic?

A: You can find further reading on this topic in the following resources:

• "Algebra and Trigonometry" by Michael Sullivan
• "Calculus" by Michael Spivak
• "Mathematics for the Nonmathematician" by Morris Kline

Additional Questions and Answers

Q: Can I use this method to simplify other expressions?

A: Yes, you can use this method to simplify other expressions that involve fractions and algebraic expressions.

Q: How do I know when to use this method?

A: You should use this method when you have an expression that involves fractions and algebraic expressions, and you need to simplify it.

Q: Can I use this method to solve other types of problems?

A: Yes, you can use this method to solve other types of problems that involve algebraic expressions and fractions.

Q: How do I apply this method to real-world problems?

A: You can apply this method to real-world problems by using it to model population growth and decay, and to solve problems involving fractions and algebraic expressions.

Conclusion

In conclusion, we have answered some frequently asked questions about simplifying the expression $\frac{x}{x+1} - 1$ given $x = \frac{1}{6}$. We have provided step-by-step solutions and explanations to help you understand the process. We hope this article has been helpful in answering your questions and providing you with a better understanding of the topic.