Which Expression Is Equal To $\frac{2x}{x-2}-\frac{x+3}{x+5}$?A. $\frac{x^2+9x+6}{(x-2)(x+5)}$B. $\frac{3x^2+11x+6}{(x-2)(x+5)}$C. $\frac{x^2+11x-6}{(x-2)(x+5)}$D. $\frac{x^2+3x+10}{(x-2)(x+5)}$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression 2xx−2−x+3x+5\frac{2x}{x-2}-\frac{x+3}{x+5}. We will break down the steps involved in simplifying this expression and provide a clear explanation of the process.

Understanding the Expression

The given expression is a combination of two fractions, 2xx−2\frac{2x}{x-2} and x+3x+5\frac{x+3}{x+5}. To simplify this expression, we need to find a common denominator and then combine the fractions.

Finding a Common Denominator

To find a common denominator, we need to identify the denominators of the two fractions. In this case, the denominators are (x−2)(x-2) and (x+5)(x+5). The least common multiple (LCM) of these two expressions is (x−2)(x+5)(x-2)(x+5).

Simplifying the Expression

Now that we have a common denominator, we can rewrite the expression as a single fraction:

2xx−2−x+3x+5=2x(x+5)−(x+3)(x−2)(x−2)(x+5)\frac{2x}{x-2}-\frac{x+3}{x+5} = \frac{2x(x+5) - (x+3)(x-2)}{(x-2)(x+5)}

Expanding the Numerator

To simplify the numerator, we need to expand the expressions 2x(x+5)2x(x+5) and (x+3)(x−2)(x+3)(x-2).

2x(x+5)=2x2+10x2x(x+5) = 2x^2 + 10x

(x+3)(x−2)=x2−2x+3x−6=x2+x−6(x+3)(x-2) = x^2 - 2x + 3x - 6 = x^2 + x - 6

Simplifying the Numerator (Continued)

Now that we have expanded the expressions, we can simplify the numerator by combining like terms:

2x2+10x−(x2+x−6)=2x2+10x−x2−x+6=x2+9x+62x^2 + 10x - (x^2 + x - 6) = 2x^2 + 10x - x^2 - x + 6 = x^2 + 9x + 6

Simplifying the Expression (Continued)

Now that we have simplified the numerator, we can rewrite the expression as:

2xx−2−x+3x+5=x2+9x+6(x−2)(x+5)\frac{2x}{x-2}-\frac{x+3}{x+5} = \frac{x^2 + 9x + 6}{(x-2)(x+5)}

Conclusion

In this article, we have simplified the algebraic expression 2xx−2−x+3x+5\frac{2x}{x-2}-\frac{x+3}{x+5} by finding a common denominator and combining the fractions. We have also expanded and simplified the numerator to arrive at the final expression. The correct answer is:

x2+9x+6(x−2)(x+5)\boxed{\frac{x^2 + 9x + 6}{(x-2)(x+5)}}

Discussion

The process of simplifying algebraic expressions is a crucial skill in mathematics, and it requires a clear understanding of the concepts involved. In this article, we have provided a step-by-step guide to simplifying the given expression, and we have also discussed the importance of finding a common denominator and combining fractions.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes to avoid. These include:

  • Failing to find a common denominator
  • Not combining like terms
  • Not simplifying the numerator
  • Not checking the final expression for errors

Tips for Simplifying Algebraic Expressions

To simplify algebraic expressions effectively, follow these tips:

  • Identify the denominators and find a common denominator
  • Combine fractions by adding or subtracting the numerators
  • Expand and simplify the numerator
  • Check the final expression for errors

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a clear understanding of the concepts involved. By following the steps outlined in this article, you can simplify complex expressions and arrive at the correct answer. Remember to find a common denominator, combine fractions, expand and simplify the numerator, and check the final expression for errors.

Final Answer

The final answer is x2+9x+6(x−2)(x+5)\boxed{\frac{x^2 + 9x + 6}{(x-2)(x+5)}}.

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Introduction

In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the given expression 2xx−2−x+3x+5\frac{2x}{x-2}-\frac{x+3}{x+5}. We provided a step-by-step guide to simplifying this expression and discussed the importance of finding a common denominator and combining fractions. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to identify the denominators and find a common denominator.

Q: How do I find a common denominator?

A: To find a common denominator, you need to identify the least common multiple (LCM) of the denominators. In the case of the expression 2xx−2−x+3x+5\frac{2x}{x-2}-\frac{x+3}{x+5}, the LCM is (x−2)(x+5)(x-2)(x+5).

Q: What is the next step after finding a common denominator?

A: After finding a common denominator, you need to rewrite the expression as a single fraction by combining the fractions.

Q: How do I combine fractions?

A: To combine fractions, you need to add or subtract the numerators while keeping the common denominator.

Q: What is the importance of expanding and simplifying the numerator?

A: Expanding and simplifying the numerator is crucial in simplifying algebraic expressions. It helps to eliminate any unnecessary terms and arrive at the final expression.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Failing to find a common denominator
  • Not combining like terms
  • Not simplifying the numerator
  • Not checking the final expression for errors

Q: How can I simplify complex algebraic expressions?

A: To simplify complex algebraic expressions, follow these steps:

  1. Identify the denominators and find a common denominator.
  2. Rewrite the expression as a single fraction by combining the fractions.
  3. Expand and simplify the numerator.
  4. Check the final expression for errors.

Q: What is the final answer to the expression 2xx−2−x+3x+5\frac{2x}{x-2}-\frac{x+3}{x+5}?

A: The final answer to the expression 2xx−2−x+3x+5\frac{2x}{x-2}-\frac{x+3}{x+5} is x2+9x+6(x−2)(x+5)\boxed{\frac{x^2 + 9x + 6}{(x-2)(x+5)}}.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a clear understanding of the concepts involved. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex expressions and arrive at the correct answer.

Final Tips

  • Practice simplifying algebraic expressions regularly to improve your skills.
  • Use online resources and tools to help you simplify complex expressions.
  • Check your work carefully to avoid errors.

Common Algebraic Expressions

Here are some common algebraic expressions that you may encounter:

  • x+2x−3−x−1x+2\frac{x+2}{x-3} - \frac{x-1}{x+2}
  • 2xx−1+x+2x+1\frac{2x}{x-1} + \frac{x+2}{x+1}
  • x−2x+1−x+1x−2\frac{x-2}{x+1} - \frac{x+1}{x-2}

Simplifying Algebraic Expressions: A Summary

Simplifying algebraic expressions is a crucial skill in mathematics that requires a clear understanding of the concepts involved. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex expressions and arrive at the correct answer.

Final Answer

The final answer is x2+9x+6(x−2)(x+5)\boxed{\frac{x^2 + 9x + 6}{(x-2)(x+5)}}.