Divide.$\[ \frac{x-2}{x^2+3x+2} \div \frac{3x-6}{x^2-2x-3} \\]

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\frac{x-2}{x^2+3x+2} \div \frac{3x-6}{x^2-2x-3} }$

Introduction

When dealing with division of fractions, it's essential to understand the concept of inverting the second fraction and multiplying instead. This process is crucial in simplifying complex expressions and solving mathematical problems. In this article, we will delve into the world of division of fractions, focusing on the given problem: x−2x2+3x+2÷3x−6x2−2x−3\frac{x-2}{x^2+3x+2} \div \frac{3x-6}{x^2-2x-3}. We will break down the solution step by step, providing a clear understanding of the process and the final result.

Understanding the Concept of Division of Fractions

Division of fractions involves inverting the second fraction and multiplying instead. This concept is based on the property that division is equivalent to multiplication by the reciprocal of the divisor. In mathematical terms, this can be represented as:

ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

This property is the foundation of solving division of fractions problems, and it's essential to understand and apply it correctly.

Simplifying the Given Problem

To simplify the given problem, we will apply the concept of inverting the second fraction and multiplying instead. The given problem is:

x−2x2+3x+2÷3x−6x2−2x−3\frac{x-2}{x^2+3x+2} \div \frac{3x-6}{x^2-2x-3}

We will start by inverting the second fraction and multiplying instead:

x−2x2+3x+2×x2−2x−33x−6\frac{x-2}{x^2+3x+2} \times \frac{x^2-2x-3}{3x-6}

Factoring the Numerators and Denominators

To simplify the expression further, we will factor the numerators and denominators:

x−2(x+1)(x+2)×(x−3)(x+1)3(x−2)\frac{x-2}{(x+1)(x+2)} \times \frac{(x-3)(x+1)}{3(x-2)}

Canceling Common Factors

Now that we have factored the numerators and denominators, we can cancel common factors:

(x−2)(x−3)(x+1)3(x+1)(x+2)\frac{(x-2)(x-3)(x+1)}{3(x+1)(x+2)}

Canceling the Common Factor (x+1)

We can cancel the common factor (x+1) from the numerator and denominator:

(x−2)(x−3)3(x+2)\frac{(x-2)(x-3)}{3(x+2)}

Final Result

The final result of the division of fractions problem is:

(x−2)(x−3)3(x+2)\frac{(x-2)(x-3)}{3(x+2)}

This is the simplified form of the given problem, and it's essential to understand the steps involved in simplifying the expression.

Conclusion

In conclusion, division of fractions involves inverting the second fraction and multiplying instead. This concept is based on the property that division is equivalent to multiplication by the reciprocal of the divisor. By applying this concept and simplifying the given problem, we arrived at the final result: (x−2)(x−3)3(x+2)\frac{(x-2)(x-3)}{3(x+2)}. This article has provided a clear understanding of the process and the final result, making it an essential resource for anyone dealing with division of fractions.

Frequently Asked Questions

  • Q: What is the concept of division of fractions? A: The concept of division of fractions involves inverting the second fraction and multiplying instead.
  • Q: How do I simplify a division of fractions problem? A: To simplify a division of fractions problem, you need to apply the concept of inverting the second fraction and multiplying instead.
  • Q: What is the final result of the given problem? A: The final result of the given problem is (x−2)(x−3)3(x+2)\frac{(x-2)(x-3)}{3(x+2)}.

Additional Resources

  • For more information on division of fractions, please refer to the following resources:
  • Khan Academy: Division of Fractions
  • Mathway: Division of Fractions
  • Wolfram Alpha: Division of Fractions

Final Thoughts

Division of fractions is a fundamental concept in mathematics, and it's essential to understand and apply it correctly. By following the steps outlined in this article, you can simplify complex expressions and solve mathematical problems with ease. Remember to always apply the concept of inverting the second fraction and multiplying instead, and you will be well on your way to mastering division of fractions.

Introduction

Division of fractions is a fundamental concept in mathematics, and it's essential to understand and apply it correctly. In this article, we will address some of the most frequently asked questions about division of fractions, providing clear and concise answers to help you better understand the concept.

Q&A

Q: What is the concept of division of fractions?

A: The concept of division of fractions involves inverting the second fraction and multiplying instead. This concept is based on the property that division is equivalent to multiplication by the reciprocal of the divisor.

Q: How do I simplify a division of fractions problem?

A: To simplify a division of fractions problem, you need to apply the concept of inverting the second fraction and multiplying instead. This involves factoring the numerators and denominators, canceling common factors, and simplifying the expression.

Q: What is the difference between division and multiplication of fractions?

A: The main difference between division and multiplication of fractions is the order in which you perform the operations. When dividing fractions, you invert the second fraction and multiply instead. When multiplying fractions, you multiply the numerators and denominators separately.

Q: Can I cancel common factors in a division of fractions problem?

A: Yes, you can cancel common factors in a division of fractions problem. However, you need to make sure that the common factors are present in both the numerator and denominator.

Q: How do I handle negative numbers in a division of fractions problem?

A: When handling negative numbers in a division of fractions problem, you need to follow the rules of arithmetic. If the numerator and denominator have the same sign, the result is positive. If the numerator and denominator have different signs, the result is negative.

Q: Can I use a calculator to simplify a division of fractions problem?

A: Yes, you can use a calculator to simplify a division of fractions problem. However, it's essential to understand the concept and the steps involved in simplifying the expression.

Q: What is the final result of the given problem?

A: The final result of the given problem is (x−2)(x−3)3(x+2)\frac{(x-2)(x-3)}{3(x+2)}.

Q: How do I check my work when simplifying a division of fractions problem?

A: To check your work when simplifying a division of fractions problem, you need to multiply the numerator and denominator by the reciprocal of the divisor. If the result is the same as the original expression, then your work is correct.

Q: Can I use a graphing calculator to visualize a division of fractions problem?

A: Yes, you can use a graphing calculator to visualize a division of fractions problem. However, it's essential to understand the concept and the steps involved in simplifying the expression.

Conclusion

Division of fractions is a fundamental concept in mathematics, and it's essential to understand and apply it correctly. By following the steps outlined in this article, you can simplify complex expressions and solve mathematical problems with ease. Remember to always apply the concept of inverting the second fraction and multiplying instead, and you will be well on your way to mastering division of fractions.

Additional Resources

  • For more information on division of fractions, please refer to the following resources:
  • Khan Academy: Division of Fractions
  • Mathway: Division of Fractions
  • Wolfram Alpha: Division of Fractions

Final Thoughts

Division of fractions is a fundamental concept in mathematics, and it's essential to understand and apply it correctly. By following the steps outlined in this article, you can simplify complex expressions and solve mathematical problems with ease. Remember to always apply the concept of inverting the second fraction and multiplying instead, and you will be well on your way to mastering division of fractions.