Divide:$\frac{6x - 5}{2x + 18} \div \frac{2x - 12}{x + 9}$Simplify The Answer.

Introduction

In mathematics, dividing complex fractions can be a daunting task, especially when dealing with variables and expressions. However, with the right approach and techniques, it can be simplified to a manageable form. In this article, we will explore the process of dividing complex fractions, focusing on the given problem: $\frac{6x - 5}{2x + 18} \div \frac{2x - 12}{x + 9}$. We will break down the steps involved in simplifying this expression and provide a clear understanding of the underlying concepts.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given problem, we have two complex fractions: $\frac{6x - 5}{2x + 18}$ and $\frac{2x - 12}{x + 9}$. To simplify the division of these fractions, we need to understand the concept of inverting and multiplying.

Inverting and Multiplying

When dividing two fractions, we can invert the second fraction (i.e., flip the numerator and denominator) and then multiply the two fractions. This process is based on the following property:

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

Applying the Inverting and Multiplying Technique

Now, let's apply this technique to the given problem:

$$\frac{6x - 5}{2x + 18} \div \frac{2x - 12}{x + 9} = \frac{6x - 5}{2x + 18} \times \frac{x + 9}{2x - 12}$$

Simplifying the Expression

To simplify the expression, we need to multiply the numerators and denominators separately:

$$\frac{(6x - 5)(x + 9)}{(2x + 18)(2x - 12)}$$

Expanding the Numerator and Denominator

Now, let's expand the numerator and denominator:

$$\frac{6x^2 + 54x - 5x - 45}{4x^2 - 24x - 36x + 216}$$

Combining Like Terms

We can combine like terms in the numerator and denominator:

$$\frac{6x^2 + 49x - 45}{4x^2 - 60x + 216}$$

Factoring the Numerator and Denominator

To simplify the expression further, we can factor the numerator and denominator:

$$\frac{(3x - 5)(2x + 9)}{(2x - 18)(2x - 12)}$$

Canceling Common Factors

We can cancel common factors between the numerator and denominator:

$$\frac{3x - 5}{2x - 18}$$

Conclusion

In this article, we have explored the process of dividing complex fractions, focusing on the given problem: $\frac{6x - 5}{2x + 18} \div \frac{2x - 12}{x + 9}$. We have broken down the steps involved in simplifying this expression, including inverting and multiplying, expanding the numerator and denominator, combining like terms, factoring, and canceling common factors. By following these steps, we have simplified the expression to its final form: $\frac{3x - 5}{2x - 18}$. This example demonstrates the importance of understanding complex fractions and the techniques involved in simplifying them.

Final Answer

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Introduction

In our previous article, we explored the process of dividing complex fractions, focusing on the given problem: $\frac{6x - 5}{2x + 18} \div \frac{2x - 12}{x + 9}$. We broke down the steps involved in simplifying this expression, including inverting and multiplying, expanding the numerator and denominator, combining like terms, factoring, and canceling common factors. In this article, we will address some common questions and concerns related to simplifying complex fractions.

Q&A

Q: What is the difference between inverting and multiplying and dividing fractions?

A: Inverting and multiplying is a technique used to simplify the division of fractions. When dividing two fractions, we can invert the second fraction (i.e., flip the numerator and denominator) and then multiply the two fractions. This process is based on the following property:

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

Q: How do I know when to invert and multiply?

A: You should invert and multiply when dividing two fractions. This technique is used to simplify the division of fractions and make it easier to work with.

Q: What is the purpose of expanding the numerator and denominator?

A: Expanding the numerator and denominator is a step in simplifying the expression. It involves multiplying out the terms in the numerator and denominator to get a simpler expression.

Q: How do I combine like terms?

A: Combining like terms involves adding or subtracting terms that have the same variable and exponent. For example, $2x + 3x$ can be combined to get $5x$.

Q: What is factoring?

A: Factoring involves expressing an expression as a product of simpler expressions. For example, $6x^2 + 9x$ can be factored as $3x(2x + 3)$.

Q: How do I cancel common factors?

A: Canceling common factors involves dividing out terms that appear in both the numerator and denominator. For example, if we have $\frac{6x}{2x}$, we can cancel the common factor of $2x$ to get $3$.

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

A: Some common mistakes to avoid when simplifying complex fractions include:

• Not inverting and multiplying when dividing fractions
• Not expanding the numerator and denominator
• Not combining like terms
• Not factoring
• Not canceling common factors

Q: How do I know when I have simplified the expression enough?

A: You have simplified the expression enough when it is in its simplest form, meaning that there are no more like terms to combine, no more common factors to cancel, and no more factoring to do.

Conclusion

In this article, we have addressed some common questions and concerns related to simplifying complex fractions. We have provided explanations and examples to help clarify the concepts and techniques involved in simplifying complex fractions. By following these steps and avoiding common mistakes, you can simplify complex fractions with confidence.

Final Tips

• Always invert and multiply when dividing fractions
• Expand the numerator and denominator to get a simpler expression
• Combine like terms to simplify the expression
• Factor to express the expression as a product of simpler expressions
• Cancel common factors to simplify the expression
• Check your work to make sure you have simplified the expression enough

Final Answer

The final answer is $\boxed{\frac{3x - 5}{2x - 18}}$.