What Is The Sum Of The Rational Expressions Below? X X − 2 + 7 X X + 3 \frac{x}{x-2}+\frac{7x}{x+3} X − 2 X ​ + X + 3 7 X ​ A. 8 Π 2 X + 3 \frac{8\pi}{2x+3} 2 X + 3 8 Π ​ B. 8 Π 8 Π \frac{8\pi}{8\pi} 8 Π 8 Π ​ C. 8 X 3 − 9 X X 2 + 3 X − 10 \frac{8x^3-9x}{x^2+3x-10} X 2 + 3 X − 10 8 X 3 − 9 X ​ D. 8 X 3 − 9 X 2 X + 3 \frac{8x^3-9x}{2x+3} 2 X + 3 8 X 3 − 9 X ​

Introduction

Rational expressions are a fundamental concept in algebra, and understanding how to add them is crucial for solving various mathematical problems. In this article, we will explore the process of adding rational expressions, focusing on the given problem: $\frac{x}{x-2}+\frac{7x}{x+3}$. We will break down the solution step by step, explaining each concept and providing examples to reinforce our understanding.

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 added, subtracted, multiplied, and divided, just like regular fractions. However, when adding or subtracting rational expressions, we must have a common denominator.

Adding Rational Expressions

To add rational expressions, we follow these steps:

1. Find the Least Common Denominator (LCD): The LCD is the smallest expression that both denominators can divide into evenly. In this case, the denominators are $(x-2)$ and $(x+3)$.
2. Multiply Each Expression by the Necessary Factor: To make the denominators the same, we multiply each expression by the necessary factor. For the first expression, we multiply by $\frac{x+3}{x+3}$, and for the second expression, we multiply by $\frac{x-2}{x-2}$.
3. Add the Numerators: Once we have the same denominator, we can add the numerators.
4. Simplify the Expression: Finally, we simplify the expression by combining like terms.

Solving the Given Problem

Now, let's apply these steps to the given problem: $\frac{x}{x-2}+\frac{7x}{x+3}$.

Step 1: Find the Least Common Denominator (LCD)

The LCD of $(x-2)$ and $(x+3)$ is $(x-2)(x+3)$.

Step 2: Multiply Each Expression by the Necessary Factor

To make the denominators the same, we multiply the first expression by $\frac{x+3}{x+3}$ and the second expression by $\frac{x-2}{x-2}$.

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

$\frac{7x}{x+3} \cdot \frac{x-2}{x-2} = \frac{7x(x-2)}{(x+3)(x-2)}$

Step 3: Add the Numerators

Now that we have the same denominator, we can add the numerators.

$\frac{x(x+3)}{(x-2)(x+3)} + \frac{7x(x-2)}{(x+3)(x-2)} = \frac{x(x+3) + 7x(x-2)}{(x-2)(x+3)}$

Step 4: Simplify the Expression

Finally, we simplify the expression by combining like terms.

$\frac{x(x+3) + 7x(x-2)}{(x-2)(x+3)} = \frac{x^2 + 3x + 7x^2 - 14x}{(x-2)(x+3)}$

$\frac{x^2 + 3x + 7x^2 - 14x}{(x-2)(x+3)} = \frac{8x^2 - 11x}{(x-2)(x+3)}$

$\frac{8x^2 - 11x}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

$\frac{8x(x-11/8)}{(x-2)(x+3)} = \frac{8x(x-11/8)}{(x-2)(x+3)}$

Q&A: Adding Rational Expressions

Q: What is the least common denominator (LCD) of two rational expressions? A: The least common denominator (LCD) is the smallest expression that both denominators can divide into evenly.

Q: How do I find the LCD of two rational expressions? A: To find the LCD, you need to factor both denominators and then multiply the factors together.

Q: What is the process of adding rational expressions? A: The process of adding rational expressions involves finding the least common denominator (LCD), multiplying each expression by the necessary factor, adding the numerators, and simplifying the expression.

Q: What is the difference between adding and subtracting rational expressions? A: The main difference between adding and subtracting rational expressions is that when subtracting, you need to change the sign of the second expression before adding.

Q: Can I add rational expressions with different signs? A: Yes, you can add rational expressions with different signs. However, you need to follow the same process as adding rational expressions with the same sign.

Q: What is the final answer to the given problem: $\frac{x}{x-2}+\frac{7x}{x+3}$? A: The final answer to the given problem is $\frac{8x^3-9x}{x^2+3x-10}$.

Q: How do I simplify a rational expression? A: To simplify a rational expression, you need to factor the numerator and denominator, cancel out any common factors, and then simplify the expression.

Q: What is the importance of finding the least common denominator (LCD) when adding rational expressions? A: Finding the least common denominator (LCD) is crucial when adding rational expressions because it allows you to combine the expressions into a single expression.

Q: Can I add rational expressions with variables in the denominator? A: Yes, you can add rational expressions with variables in the denominator. However, you need to follow the same process as adding rational expressions with constants in the denominator.

Q: What is the final answer to the problem: $\frac{8x^3-9x}{x^2+3x-10}$? A: The final answer to the problem is $\frac{8x^3-9x}{x^2+3x-10}$.

Conclusion

Adding rational expressions is a crucial concept in algebra, and understanding how to do it is essential for solving various mathematical problems. By following the steps outlined in this article, you can add rational expressions with ease. Remember to find the least common denominator (LCD), multiply each expression by the necessary factor, add the numerators, and simplify the expression. With practice, you will become proficient in adding rational expressions and be able to solve complex problems with confidence.

Additional Resources

• Khan Academy: Adding and Subtracting Rational Expressions
• Mathway: Adding Rational Expressions
• Purplemath: Adding and Subtracting Rational Expressions

Practice Problems

1. Add the rational expressions: $\frac{x}{x+1}+\frac{2x}{x-2}$
2. Subtract the rational expressions: $\frac{3x}{x+2}-\frac{2x}{x-1}$
3. Add the rational expressions: $\frac{x^2}{x+1}+\frac{2x}{x-2}$

Answer Key

1. $\frac{x^2+3x-2}{x^2-3x-2}$
2. $\frac{5x^2-7x-2}{x^2+3x-2}$
3. $\frac{x^3+3x^2-2x}{x^2-3x-2}$