What Is The Sum Of The Rational Expressions Below? X 5 + 2 3 X \frac{x}{5} + \frac{2}{3x} 5 X ​ + 3 X 2 ​ A. X + 2 15 X \frac{x+2}{15x} 15 X X + 2 ​ B. X + 2 3 X + 5 \frac{x+2}{3x+5} 3 X + 5 X + 2 ​ C. 3 X 2 + 10 15 X \frac{3x^2+10}{15x} 15 X 3 X 2 + 10 ​ D. 3 X 2 + 10 3 X + 5 \frac{3x^2+10}{3x+5} 3 X + 5 3 X 2 + 10 ​

Introduction

Rational expressions are a fundamental concept in algebra, and understanding how to add and subtract them is crucial for solving various mathematical problems. In this article, we will explore the sum of rational expressions, focusing on the given problem: $\frac{x}{5} + \frac{2}{3x}$. We will break down the solution step by step, explaining each concept and providing examples to illustrate the process.

Understanding Rational Expressions

A rational expression is a fraction that contains variables, constants, or a combination of both in the numerator and denominator. Rational expressions can be added or subtracted by finding a common denominator and combining the numerators. The common denominator is the least common multiple (LCM) of the denominators of the rational expressions.

Step 1: Find the Common Denominator

To add the rational expressions $\frac{x}{5}$ and $\frac{2}{3x}$, we need to find the least common multiple (LCM) of the denominators, which are 5 and $3x$. The LCM of 5 and $3x$ is $15x$.

Step 2: Rewrite the Rational Expressions

Now that we have the common denominator, we can rewrite each rational expression with the common denominator.

$\frac{x}{5} = \frac{x \cdot 3x}{5 \cdot 3x} = \frac{3x^2}{15x}$

$\frac{2}{3x} = \frac{2 \cdot 5}{3x \cdot 5} = \frac{10}{15x}$

Step 3: Add the Rational Expressions

Now that we have rewritten the rational expressions with the common denominator, we can add them.

$\frac{3x^2}{15x} + \frac{10}{15x} = \frac{3x^2 + 10}{15x}$

Conclusion

In conclusion, the sum of the rational expressions $\frac{x}{5} + \frac{2}{3x}$ is $\frac{3x^2 + 10}{15x}$. This solution demonstrates the importance of finding the common denominator and rewriting the rational expressions before adding them.

Comparison with Answer Choices

Now that we have found the sum of the rational expressions, let's compare it with the answer choices.

A. $\frac{x+2}{15x}$

B. $\frac{x+2}{3x+5}$

C. $\frac{3x^2+10}{15x}$

D. $\frac{3x^2+10}{3x+5}$

The correct answer is C. $\frac{3x^2+10}{15x}$.

Tips and Tricks

When adding rational expressions, remember to find the least common multiple (LCM) of the denominators and rewrite each rational expression with the common denominator. This will ensure that you are adding the numerators correctly.

Real-World Applications

Rational expressions are used in various real-world applications, such as:

• Finance: Rational expressions are used to calculate interest rates, investment returns, and other financial metrics.
• Science: Rational expressions are used to model physical systems, such as the motion of objects and the behavior of electrical circuits.
• Engineering: Rational expressions are used to design and optimize systems, such as bridges, buildings, and electronic devices.

Conclusion

In conclusion, the sum of rational expressions is a fundamental concept in algebra that requires finding the common denominator and rewriting the rational expressions before adding them. By following these steps, you can solve various mathematical problems and apply rational expressions to real-world applications.

Final Answer

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Frequently Asked Questions

In this article, we will address some of the most common questions related to rational expressions.

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables, constants, or a combination of both in the numerator and denominator.

Q: How do I add rational expressions?

A: To add rational expressions, you need to find the least common multiple (LCM) of the denominators and rewrite each rational expression with the common denominator. Then, you can add the numerators.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 5 and 3x is 15x.

Q: How do I find the LCM of two or more numbers?

A: To find the LCM of two or more numbers, you can list the multiples of each number and find the smallest multiple that they all have in common.

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

A: A rational number is a number that can be expressed as the ratio of two integers, such as 3/4. A rational expression is a fraction that contains variables, constants, or a combination of both in the numerator and denominator.

Q: Can I simplify a rational expression?

A: Yes, you can simplify a rational expression by canceling out any common factors in the numerator and denominator.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator and cancel out any common factors.

Q: What is the final answer to the problem $\frac{x}{5} + \frac{2}{3x}$?

A: The final answer to the problem $\frac{x}{5} + \frac{2}{3x}$ is $\frac{3x^2 + 10}{15x}$.

Q: Can I use rational expressions in real-world applications?

A: Yes, rational expressions are used in various real-world applications, such as finance, science, and engineering.

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

A: Some examples of real-world applications of rational expressions include:

• Finance: Rational expressions are used to calculate interest rates, investment returns, and other financial metrics.
• Science: Rational expressions are used to model physical systems, such as the motion of objects and the behavior of electrical circuits.
• Engineering: Rational expressions are used to design and optimize systems, such as bridges, buildings, and electronic devices.

Conclusion

In conclusion, rational expressions are a fundamental concept in algebra that have numerous real-world applications. By understanding how to add and simplify rational expressions, you can solve various mathematical problems and apply rational expressions to real-world applications.

Final Answer

The final answer is $\boxed{\frac{3x^2+10}{15x}}$.