Expressed As A Single Fraction, $\frac{3}{x-1}-\frac{2}{x}$ Is Equivalent To:A. $\frac{1}{x(x-1)}$B. $\frac{x-2}{x(x-1)}$C. $\frac{x+2}{x(x-1)}$D. $\frac{3x-2}{x(x-1)}$

Feb 26, 2025 by ADMIN 169 views

**Expressed as a Single Fraction: Simplifying Rational Expressions**

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 simplify the rational expression $\frac{3}{x-1}-\frac{2}{x}$ and express it as a single fraction. We will also examine the different options provided and determine which one is equivalent to the given expression.

Understanding Rational Expressions

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by canceling out common factors in the numerator and denominator. In this case, we have two rational expressions: $\frac{3}{x-1}$ and $\frac{2}{x}$ .

Step 1: Find a Common Denominator

To simplify the given expression, we need to find a common denominator for both fractions. The least common multiple (LCM) of $(x-1)$ and $x$ is $x(x-1)$ . Therefore, we can rewrite the expression as:

\frac{3}{x-1}-\frac{2}{x}=\frac{3x}{x(x-1)}-\frac{2(x-1)}{x(x-1)}

Step 2: Combine the Fractions

Now that we have a common denominator, we can combine the fractions by adding or subtracting the numerators:

\frac{3x}{x(x-1)}-\frac{2(x-1)}{x(x-1)}=\frac{3x-2(x-1)}{x(x-1)}

Step 3: Simplify the Numerator

We can simplify the numerator by distributing the negative sign and combining like terms:

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

Conclusion

Based on our calculations, we have simplified the rational expression $\frac{3}{x-1}-\frac{2}{x}$ and expressed it as a single fraction: $\frac{x+2}{x(x-1)}$ . This is equivalent to option C.

Comparison of Options

Let's compare our result with the other options:

Option A: $\frac{1}{x(x-1)}$ is not equivalent to the given expression.
Option B: $\frac{x-2}{x(x-1)}$ is not equivalent to the given expression.
Option D: $\frac{3x-2}{x(x-1)}$ is not equivalent to the given expression.

Final Answer

The final answer is $\boxed{\frac{x+2}{x(x-1)}}$ .

Additional Tips and Tricks

When simplifying rational expressions, it's essential to find a common denominator and combine the fractions. You can also use the distributive property to simplify the numerator. Remember to check your work by plugging in values for the variables to ensure that the expression is equivalent to the original one.

Common Mistakes to Avoid

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

Failing to find a common denominator
Not combining the fractions correctly
Not simplifying the numerator properly
Not checking your work

By following these tips and avoiding common mistakes, you can simplify rational expressions with confidence and accuracy.

Real-World Applications

Rational expressions have numerous real-world applications in fields such as engineering, economics, and physics. For example, rational expressions can be used to model population growth, electrical circuits, and mechanical systems.

Conclusion

Introduction

In our previous article, we explored how to simplify the rational expression $\frac{3}{x-1}-\frac{2}{x}$ and express it as a single fraction. 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 and/or constants in the numerator and/or denominator.

A: Rational expressions can be simplified by canceling out common factors in the numerator and denominator.

Q: How do I find a common denominator?

To find a common denominator, you need to identify the least common multiple (LCM) of the denominators. For example, if you have two fractions with denominators of $x$ and $x-1$ , the LCM is $x(x-1)$ .

A: Once you have found the common denominator, you can rewrite the fractions with the common denominator and then combine them.

Q: How do I combine fractions?

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

A: For example, if you have two fractions with the same denominator, you can add or subtract the numerators to get a new fraction.

Q: What is the distributive property?

The distributive property is a rule that allows you to multiply a single term by multiple terms.

A: For example, if you have the expression $3(x-1)$ , you can use the distributive property to expand it to $3x-3$ .

Q: How do I simplify the numerator?

To simplify the numerator, you need to combine like terms and eliminate any common factors.

A: For example, if you have the expression $3x-2(x-1)$ , you can simplify it to $3x-2x+2$ .

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

Some common mistakes to avoid when simplifying rational expressions include:

Failing to find a common denominator
Not combining the fractions correctly
Not simplifying the numerator properly
Not checking your work

A: By following these tips and avoiding common mistakes, you can simplify rational expressions with confidence and accuracy.

Q: What are some real-world applications of rational expressions?

Rational expressions have numerous real-world applications in fields such as engineering, economics, and physics.

A: For example, rational expressions can be used to model population growth, electrical circuits, and mechanical systems.

Q: How do I check my work when simplifying rational expressions?

To check your work, you can plug in values for the variables to ensure that the expression is equivalent to the original one.

A: For example, if you have the expression $\frac{x+2}{x(x-1)}$ , you can plug in $x=1$ to get $\frac{3}{2}$ , which is equivalent to the original expression.

Conclusion

In conclusion, simplifying rational expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article and avoiding common mistakes, you can simplify rational expressions with confidence and accuracy. Remember to find a common denominator, combine the fractions, and simplify the numerator to ensure that your result is equivalent to the original expression.