Simplify: \frac{x^2 + 5x + 4}{3x} \div \left(\frac{x^2 - 16}{x^2 + 3x} \cdot \frac{x + 3}{x - 4}\right ]

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Introduction to Simplifying Complex Fractions

Simplifying complex fractions is a crucial skill in mathematics, particularly in algebra and calculus. It involves breaking down complex expressions into simpler ones, making it easier to solve equations and manipulate variables. In this article, we will focus on simplifying a specific complex fraction, $\frac{x^2 + 5x + 4}{3x} \div \left(\frac{x^2 - 16}{x^2 + 3x} \cdot \frac{x + 3}{x - 4}\right)$.

Understanding the Structure of the Complex Fraction

To simplify the given complex fraction, we need to understand its structure. The complex fraction is divided by another complex fraction, which is a product of two fractions. The first fraction in the numerator is $\frac{x^2 + 5x + 4}{3x}$, and the second fraction in the denominator is $\left(\frac{x^2 - 16}{x^2 + 3x} \cdot \frac{x + 3}{x - 4}\right)$. To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Simplifying the First Fraction in the Numerator

The first fraction in the numerator is $\frac{x^2 + 5x + 4}{3x}$. To simplify this fraction, we can factor the numerator:

$$\frac{x^2 + 5x + 4}{3x} = \frac{(x + 4)(x + 1)}{3x}$$

Simplifying the Second Fraction in the Denominator

The second fraction in the denominator is $\left(\frac{x^2 - 16}{x^2 + 3x} \cdot \frac{x + 3}{x - 4}\right)$. To simplify this fraction, we can factor the numerator and denominator:

$$\frac{x^2 - 16}{x^2 + 3x} = \frac{(x + 4)(x - 4)}{(x + 3)(x - 4)}$$

$$\frac{x + 3}{x - 4} = \frac{x + 3}{x - 4}$$

Combining the Simplified Fractions

Now that we have simplified the first fraction in the numerator and the second fraction in the denominator, we can combine them:

$$\frac{(x + 4)(x + 1)}{3x} \div \left(\frac{(x + 4)(x - 4)}{(x + 3)(x - 4)} \cdot \frac{x + 3}{x - 4}\right)$$

Canceling Out Common Factors

To simplify the expression further, we can cancel out common factors between the numerator and denominator:

$$\frac{(x + 4)(x + 1)}{3x} \div \left(\frac{(x + 4)(x - 4)}{(x + 3)(x - 4)} \cdot \frac{x + 3}{x - 4}\right) = \frac{(x + 4)(x + 1)}{3x} \div \left(\frac{(x + 4)(x + 3)}{(x + 3)(x - 4)}\right)$$

Simplifying the Expression

Now that we have canceled out common factors, we can simplify the expression further:

$$\frac{(x + 4)(x + 1)}{3x} \div \left(\frac{(x + 4)(x + 3)}{(x + 3)(x - 4)}\right) = \frac{(x + 4)(x + 1)}{3x} \cdot \frac{(x - 4)}{(x + 4)}$$

Canceling Out Common Factors Again

To simplify the expression further, we can cancel out common factors again:

$$\frac{(x + 4)(x + 1)}{3x} \cdot \frac{(x - 4)}{(x + 4)} = \frac{(x + 1)(x - 4)}{3x}$$

Final Simplification

The final simplified expression is:

$$\frac{(x + 1)(x - 4)}{3x}$$

Conclusion

Simplifying complex fractions is a crucial skill in mathematics, particularly in algebra and calculus. By following the order of operations and canceling out common factors, we can simplify complex expressions and make them easier to solve equations and manipulate variables. In this article, we simplified the complex fraction $\frac{x^2 + 5x + 4}{3x} \div \left(\frac{x^2 - 16}{x^2 + 3x} \cdot \frac{x + 3}{x - 4}\right)$ to its final simplified form: $\frac{(x + 1)(x - 4)}{3x}$.

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Introduction

In our previous article, we simplified the complex fraction $\frac{x^2 + 5x + 4}{3x} \div \left(\frac{x^2 - 16}{x^2 + 3x} \cdot \frac{x + 3}{x - 4}\right)$ to its final simplified form: $\frac{(x + 1)(x - 4)}{3x}$. However, we understand that simplifying complex fractions can be a challenging task, and many readers may have questions about the process. In this article, we will address some of the most 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. In the case of the given complex fraction, $\frac{x^2 + 5x + 4}{3x} \div \left(\frac{x^2 - 16}{x^2 + 3x} \cdot \frac{x + 3}{x - 4}\right)$, the numerator and denominator both contain fractions.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between a complex fraction and a simple fraction?

A: A simple fraction is a fraction that does not contain any fractions in its numerator or denominator. For example, $\frac{2}{3}$ is a simple fraction. A complex fraction, on the other hand, contains one or more fractions in its numerator or denominator.

Q: Can I simplify a complex fraction by canceling out common factors?

A: Yes, you can simplify a complex fraction by canceling out common factors between the numerator and denominator. However, you need to be careful not to cancel out any factors that are not common to both the numerator and denominator.

Q: How do I know when to cancel out common factors?

A: To determine when to cancel out common factors, you need to look for factors that are present in both the numerator and denominator. If you find any common factors, you can cancel them out.

Q: What is the final simplified form of the complex fraction $\frac{x^2 + 5x + 4}{3x} \div \left(\frac{x^2 - 16}{x^2 + 3x} \cdot \frac{x + 3}{x - 4}\right)$?

A: The final simplified form of the complex fraction $\frac{x^2 + 5x + 4}{3x} \div \left(\frac{x^2 - 16}{x^2 + 3x} \cdot \frac{x + 3}{x - 4}\right)$ is $\frac{(x + 1)(x - 4)}{3x}$.

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

A: Yes, you can use a calculator to simplify a complex fraction. However, it's always a good idea to check your work by simplifying the fraction manually to ensure that you get the correct answer.

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

A: To determine if you have simplified a complex fraction correctly, you need to check your work by plugging in values for the variables and evaluating the expression. If the expression evaluates to the correct value, then you have simplified the complex fraction correctly.

Conclusion

Simplifying complex fractions can be a challenging task, but with practice and patience, you can master the process. By following the order of operations and canceling out common factors, you can simplify complex fractions and make them easier to solve equations and manipulate variables. We hope that this Q&A article has helped to clarify any questions you may have had about simplifying complex fractions.