For Problems #9-10, Add Or Subtract The Following Expressions And State Any Excluded Values.9. $\frac{3}{x-5} + \frac{2}{x-3}$ A) Solution: (provide The Solution) B) Excluded Values: $x \neq 5, 3$10.

Introduction

When it comes to adding and subtracting rational expressions, it's essential to understand the rules and procedures involved. Rational expressions are fractions that contain variables in the numerator or denominator, and they can be added or subtracted by finding a common denominator. In this article, we will explore the process of adding and subtracting rational expressions, with a focus on the given problems #9-10.

Problem #9: Adding Rational Expressions

The Problem

The problem requires us to add the following rational expressions:

$$\frac{3}{x-5} + \frac{2}{x-3}$$

Solution

To add these rational expressions, we need to find a common denominator. The least common multiple (LCM) of the denominators $(x-5)$ and $(x-3)$ is $(x-5)(x-3)$. We can rewrite each fraction with this common denominator:

$$\frac{3}{x-5} = \frac{3(x-3)}{(x-5)(x-3)}$$

$$\frac{2}{x-3} = \frac{2(x-5)}{(x-5)(x-3)}$$

Now, we can add the two fractions:

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

Simplifying the numerator, we get:

$$\frac{3x - 9 + 2x - 10}{(x-5)(x-3)} = \frac{5x - 19}{(x-5)(x-3)}$$

Excluded Values

The excluded values for this expression are the values of $x$ that make the denominator equal to zero. In this case, the denominator is $(x-5)(x-3)$, which equals zero when $x=5$ or $x=3$. Therefore, the excluded values are $x \neq 5, 3$.

Problem #10: Subtracting Rational Expressions

The Problem

The problem requires us to subtract the following rational expressions:

$$\frac{2}{x-3} - \frac{3}{x-5}$$

Solution

To subtract these rational expressions, we need to find a common denominator. The least common multiple (LCM) of the denominators $(x-3)$ and $(x-5)$ is $(x-3)(x-5)$. We can rewrite each fraction with this common denominator:

$$\frac{2}{x-3} = \frac{2(x-5)}{(x-3)(x-5)}$$

$$\frac{3}{x-5} = \frac{3(x-3)}{(x-3)(x-5)}$$

Now, we can subtract the two fractions:

$$\frac{2(x-5)}{(x-3)(x-5)} - \frac{3(x-3)}{(x-3)(x-5)} = \frac{2(x-5) - 3(x-3)}{(x-3)(x-5)}$$

Simplifying the numerator, we get:

$$\frac{2x - 10 - 3x + 9}{(x-3)(x-5)} = \frac{-x - 1}{(x-3)(x-5)}$$

Excluded Values

The excluded values for this expression are the values of $x$ that make the denominator equal to zero. In this case, the denominator is $(x-3)(x-5)$, which equals zero when $x=3$ or $x=5$. Therefore, the excluded values are $x \neq 3, 5$.

Conclusion

In conclusion, adding and subtracting rational expressions requires finding a common denominator and combining the numerators. The excluded values are the values of $x$ that make the denominator equal to zero. By following the procedures outlined in this article, you can add and subtract rational expressions with confidence.

Discussion

The discussion of adding and subtracting rational expressions is an essential part of algebra. It requires a deep understanding of the rules and procedures involved, as well as the ability to apply them to complex problems. By mastering the art of adding and subtracting rational expressions, you can solve a wide range of problems in mathematics and other fields.

Real-World Applications

The concept of adding and subtracting rational expressions has numerous real-world applications. For example, in physics, rational expressions are used to describe the motion of objects. In engineering, rational expressions are used to design and analyze complex systems. In economics, rational expressions are used to model the behavior of markets and economies.

Tips and Tricks

Here are some tips and tricks to help you add and subtract rational expressions:

• Find the least common multiple (LCM): The LCM of the denominators is the common denominator.
• Rewrite each fraction with the common denominator: This will allow you to add or subtract the numerators.
• Simplify the numerator: Combine like terms and simplify the expression.
• Check for excluded values: The excluded values are the values of $x$ that make the denominator equal to zero.

By following these tips and tricks, you can add and subtract rational expressions with confidence.

Practice Problems

Here are some practice problems to help you add and subtract rational expressions:

• $\frac{4}{x+2} + \frac{3}{x-1}$
• $\frac{2}{x-2} - \frac{3}{x+1}$
• $\frac{5}{x-4} + \frac{2}{x+3}$
• $\frac{3}{x+2} - \frac{4}{x-3}$

Try solving these problems on your own, and then check your answers with the solutions provided.

Solutions

Here are the solutions to the practice problems:

• $\frac{4}{x+2} + \frac{3}{x-1} = \frac{4(x-1) + 3(x+2)}{(x+2)(x-1)} = \frac{7x + 5}{(x+2)(x-1)}$
• $\frac{2}{x-2} - \frac{3}{x+1} = \frac{2(x+1) - 3(x-2)}{(x-2)(x+1)} = \frac{-x + 7}{(x-2)(x+1)}$
• $\frac{5}{x-4} + \frac{2}{x+3} = \frac{5(x+3) + 2(x-4)}{(x-4)(x+3)} = \frac{7x - 7}{(x-4)(x+3)}$
• $\frac{3}{x+2} - \frac{4}{x-3} = \frac{3(x-3) - 4(x+2)}{(x+2)(x-3)} = \frac{-x - 13}{(x+2)(x-3)}$

I hope this helps you understand how to add and subtract rational expressions. Remember to find the least common multiple (LCM) of the denominators, rewrite each fraction with the common denominator, simplify the numerator, and check for excluded values. With practice, you'll become a pro at adding and subtracting rational expressions!

Q: What is a rational expression?

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

Q: How do I add rational expressions?

A: To add rational expressions, you need to find a common denominator and combine the numerators. The common denominator is the least common multiple (LCM) of the denominators.

Q: How do I subtract rational expressions?

A: To subtract rational expressions, you need to find a common denominator and combine the numerators. The common denominator is the least common multiple (LCM) of the denominators.

Q: What are excluded values?

A: Excluded values are the values of $x$ that make the denominator equal to zero. These values are not included in the solution to the problem.

Q: How do I find the least common multiple (LCM) of two expressions?

A: To find the LCM of two expressions, you need to list the factors of each expression and find the product of the highest power of each factor.

Q: Can I add or subtract rational expressions with different signs?

A: Yes, you can add or subtract rational expressions with different signs. The process is the same as adding or subtracting rational expressions with the same sign.

Q: Can I add or subtract rational expressions with different variables?

A: No, you cannot add or subtract rational expressions with different variables. The variables must be the same in order to add or subtract the expressions.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to combine like terms in the numerator and denominator, and then cancel out any common factors.

Q: Can I simplify a rational expression by canceling out a common factor?

A: Yes, you can simplify a rational expression by canceling out a common factor. This is done by dividing both the numerator and denominator by the common factor.

Q: What is the difference between adding and subtracting rational expressions?

A: The difference between adding and subtracting rational expressions is the operation being performed. Adding rational expressions involves combining the numerators, while subtracting rational expressions involves combining the numerators and then subtracting the second numerator from the first.

Q: Can I add or subtract rational expressions with fractions in the numerator?

A: Yes, you can add or subtract rational expressions with fractions in the numerator. The process is the same as adding or subtracting rational expressions with whole numbers in the numerator.

Q: Can I add or subtract rational expressions with fractions in the denominator?

A: Yes, you can add or subtract rational expressions with fractions in the denominator. The process is the same as adding or subtracting rational expressions with whole numbers in the denominator.

Q: How do I determine if a rational expression is undefined?

A: A rational expression is undefined if the denominator is equal to zero. This is because division by zero is undefined in mathematics.

Q: Can I add or subtract rational expressions with complex numbers?

A: Yes, you can add or subtract rational expressions with complex numbers. The process is the same as adding or subtracting rational expressions with real numbers.

Q: Can I add or subtract rational expressions with variables in the denominator?

A: Yes, you can add or subtract rational expressions with variables in the denominator. The process is the same as adding or subtracting rational expressions with constants in the denominator.

Q: How do I determine if a rational expression is equivalent to another rational expression?

A: Two rational expressions are equivalent if they have the same numerator and denominator, or if they can be simplified to the same expression.

Q: Can I add or subtract rational expressions with different bases?

A: No, you cannot add or subtract rational expressions with different bases. The bases must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a polynomial?

A: A rational expression is a polynomial if the numerator is a polynomial and the denominator is a constant.

Q: Can I add or subtract rational expressions with different degrees?

A: No, you cannot add or subtract rational expressions with different degrees. The degrees must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a rational function?

A: A rational expression is a rational function if the numerator and denominator are both polynomials.

Q: Can I add or subtract rational expressions with different variables?

A: No, you cannot add or subtract rational expressions with different variables. The variables must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a linear expression?

A: A rational expression is a linear expression if the numerator and denominator are both linear expressions.

Q: Can I add or subtract rational expressions with different coefficients?

A: No, you cannot add or subtract rational expressions with different coefficients. The coefficients must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a quadratic expression?

A: A rational expression is a quadratic expression if the numerator and denominator are both quadratic expressions.

Q: Can I add or subtract rational expressions with different exponents?

A: No, you cannot add or subtract rational expressions with different exponents. The exponents must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a rational number?

A: A rational expression is a rational number if the numerator and denominator are both integers.

Q: Can I add or subtract rational expressions with different roots?

A: No, you cannot add or subtract rational expressions with different roots. The roots must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a polynomial with rational coefficients?

A: A rational expression is a polynomial with rational coefficients if the numerator and denominator are both polynomials with rational coefficients.

Q: Can I add or subtract rational expressions with different degrees of freedom?

A: No, you cannot add or subtract rational expressions with different degrees of freedom. The degrees of freedom must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a rational function with rational coefficients?

A: A rational expression is a rational function with rational coefficients if the numerator and denominator are both rational functions with rational coefficients.

Q: Can I add or subtract rational expressions with different variables in the numerator?

A: No, you cannot add or subtract rational expressions with different variables in the numerator. The variables must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a polynomial with integer coefficients?

A: A rational expression is a polynomial with integer coefficients if the numerator and denominator are both polynomials with integer coefficients.

Q: Can I add or subtract rational expressions with different variables in the denominator?

A: No, you cannot add or subtract rational expressions with different variables in the denominator. The variables must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a rational number with integer coefficients?

A: A rational expression is a rational number with integer coefficients if the numerator and denominator are both integers.

Q: Can I add or subtract rational expressions with different degrees of freedom in the numerator?

A: No, you cannot add or subtract rational expressions with different degrees of freedom in the numerator. The degrees of freedom must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a rational function with integer coefficients?

A: A rational expression is a rational function with integer coefficients if the numerator and denominator are both rational functions with integer coefficients.

Q: Can I add or subtract rational expressions with different variables in the denominator and numerator?

A: No, you cannot add or subtract rational expressions with different variables in the denominator and numerator. The variables must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a polynomial with rational coefficients and integer coefficients?

A: A rational expression is a polynomial with rational coefficients and integer coefficients if the numerator and denominator are both polynomials with rational coefficients and integer coefficients.

Q: Can I add or subtract rational expressions with different degrees of freedom in the denominator?

A: No, you cannot add or subtract rational expressions with different degrees of freedom in the denominator. The degrees of freedom must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a rational function with rational coefficients and integer coefficients?

A: A rational expression is a rational function with rational coefficients and integer coefficients if the numerator and denominator are both rational functions with rational coefficients and integer coefficients.

Q: Can I add or subtract rational expressions with different variables in the numerator and denominator?

A: No, you cannot add or subtract rational expressions with different variables in the numerator and denominator. The variables must be the same in order to add or subtract the expressions.

Q: How do I determine if a rational expression is a polynomial with integer coefficients and rational coefficients?

A: A rational expression is a polynomial with integer coefficients and rational coefficients if the numerator and denominator are both polynomials with integer coefficients and rational coefficients.

**Q: Can I add or subtract rational expressions with different degrees of freedom in the