Write The Following As A Single Rational Expression.$\frac{1}{x} + \frac{1}{2-x}$A. $\frac{-2}{x(2-x)}$B. $\frac{1}{2}$C. $\frac{2}{x(2-x)}$D. $1$

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Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore how to write the given expression 1x+12−x\frac{1}{x} + \frac{1}{2-x} as a single rational expression. We will examine each option and determine the correct answer.

Understanding the Problem

The given expression is a sum of two fractions: 1x\frac{1}{x} and 12−x\frac{1}{2-x}. To simplify this expression, we need to find a common denominator and combine the fractions.

Finding a Common Denominator

To find a common denominator, we need to identify the least common multiple (LCM) of the denominators xx and 2−x2-x. Since the LCM of xx and 2−x2-x is x(2−x)x(2-x), we can rewrite the expression as:

1x+12−x=1x⋅2−x2−x+12−x⋅xx\frac{1}{x} + \frac{1}{2-x} = \frac{1}{x} \cdot \frac{2-x}{2-x} + \frac{1}{2-x} \cdot \frac{x}{x}

Simplifying the Expression

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

1x⋅2−x2−x+12−x⋅xx=2−x+xx(2−x)\frac{1}{x} \cdot \frac{2-x}{2-x} + \frac{1}{2-x} \cdot \frac{x}{x} = \frac{2-x+x}{x(2-x)}

Evaluating the Expression

Simplifying the numerator, we get:

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

Conclusion

Based on our analysis, the correct answer is:

2x(2−x)\frac{2}{x(2-x)}

This is the simplified form of the given expression 1x+12−x\frac{1}{x} + \frac{1}{2-x}.

Comparison with Other Options

Let's compare our answer with the other options:

  • Option A: −2x(2−x)\frac{-2}{x(2-x)} is incorrect because the numerator is negative.
  • Option B: 12\frac{1}{2} is incorrect because it does not match the original expression.
  • Option D: 11 is incorrect because it does not match the original expression.

Final Thoughts

Simplifying rational expressions requires a clear understanding of the concept and the ability to apply mathematical operations. By following the steps outlined in this article, you can simplify even the most complex rational expressions. Remember to always find a common denominator and combine the fractions to get the final answer.

Common Mistakes to Avoid

When simplifying rational expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not finding a common denominator
  • Not combining the fractions
  • Not simplifying the numerator

Practice Problems

To practice simplifying rational expressions, try the following problems:

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

Conclusion

Introduction

In our previous article, we explored how to simplify rational expressions by finding a common denominator and combining the fractions. In this article, we will answer some frequently asked questions about simplifying rational expressions.

Q: What is a rational expression?

A rational expression is a fraction that contains variables in the numerator or denominator. It is a way to represent a relationship between two quantities.

Q: How do I simplify a rational expression?

To simplify a rational expression, you need to find a common denominator and combine the fractions. This involves multiplying the numerator and denominator of each fraction by the necessary factors to get a common denominator.

Q: What is a common denominator?

A common denominator is the least common multiple (LCM) of the denominators of two or more fractions. It is the smallest number that both denominators can divide into evenly.

Q: How do I find a common denominator?

To find a common denominator, you need to identify the LCM of the denominators. This can be done by listing the multiples of each denominator and finding the smallest number that appears in both lists.

Q: What if the denominators are not factorable?

If the denominators are not factorable, you can use the following methods to find a common denominator:

  • Use the prime factorization method to find the LCM of the denominators.
  • Use the greatest common divisor (GCD) method to find the LCM of the denominators.

Q: Can I simplify a rational expression with a variable in the denominator?

Yes, you can simplify a rational expression with a variable in the denominator. However, you need to be careful when simplifying expressions with variables in the denominator.

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

Some common mistakes to avoid when simplifying rational expressions include:

  • Not finding a common denominator
  • Not combining the fractions
  • Not simplifying the numerator
  • Not canceling out any common factors

Q: How do I know if a rational expression is already simplified?

A rational expression is already simplified if it cannot be simplified further by canceling out any common factors.

Q: Can I simplify a rational expression with a negative exponent?

Yes, you can simplify a rational expression with a negative exponent. However, you need to be careful when simplifying expressions with negative exponents.

Q: What is the difference between simplifying a rational expression and reducing a rational expression?

Simplifying a rational expression involves finding a common denominator and combining the fractions. Reducing a rational expression involves canceling out any common factors.

Q: Can I simplify a rational expression with a complex fraction?

Yes, you can simplify a rational expression with a complex fraction. However, you need to be careful when simplifying expressions with complex fractions.

Conclusion

Simplifying rational expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article, you can simplify even the most complex rational expressions. Remember to always find a common denominator and combine the fractions to get the final answer. With practice, you'll become proficient in simplifying rational expressions and be able to tackle even the most challenging problems.

Practice Problems

To practice simplifying rational expressions, try the following problems:

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

Additional Resources

For more information on simplifying rational expressions, check out the following resources:

  • Khan Academy: Simplifying Rational Expressions
  • Mathway: Simplifying Rational Expressions
  • Wolfram Alpha: Simplifying Rational Expressions