Solve The Following Rational Expression:${ \frac{5}{x+1} - \frac{4}{x-2} }$

Introduction

Rational expressions are a fundamental concept in algebra, and solving them is a crucial skill for any math enthusiast. In this article, we will delve into the world of rational expressions and provide a step-by-step guide on how to solve the given expression: $\frac{5}{x+1} - \frac{4}{x-2}$. We will break down the solution into manageable parts, making it easy to understand and follow.

What are Rational Expressions?

Before we dive into the solution, let's briefly discuss what rational expressions are. A rational expression is a fraction that contains variables in the numerator or denominator. It is called "rational" because it is the ratio of two polynomials. Rational expressions can be added, subtracted, multiplied, and divided, just like regular fractions.

The Given Expression

The given expression is $\frac{5}{x+1} - \frac{4}{x-2}$. Our goal is to simplify this expression and find a common denominator.

Step 1: Find a Common Denominator

To add or subtract fractions, we need to find a common denominator. In this case, the denominators are $(x+1)$ and $(x-2)$. To find a common denominator, we need to multiply both denominators by the same value.

import sympy as sp
x = sp.symbols('x')

denom1 = x + 1
denom2 = x - 2

common_denom = denom1 * denom2
print(common_denom)

The output of the code is $(x+1)(x-2)$, which is the common denominator.

Step 2: Rewrite the Expression with the Common Denominator

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

$\frac{5}{x+1} - \frac{4}{x-2} = \frac{5(x-2)}{(x+1)(x-2)} - \frac{4(x+1)}{(x+1)(x-2)}$

Step 3: Simplify the Expression

Now that we have rewritten the expression with the common denominator, we can simplify it by combining the fractions.

$\frac{5(x-2)}{(x+1)(x-2)} - \frac{4(x+1)}{(x+1)(x-2)} = \frac{5(x-2) - 4(x+1)}{(x+1)(x-2)}$

Step 4: Expand and Simplify

Now that we have simplified the expression, we can expand and simplify it further.

$\frac{5(x-2) - 4(x+1)}{(x+1)(x-2)} = \frac{5x - 10 - 4x - 4}{(x+1)(x-2)}$

Step 5: Combine Like Terms

Now that we have expanded and simplified the expression, we can combine like terms.

$\frac{5x - 10 - 4x - 4}{(x+1)(x-2)} = \frac{x - 14}{(x+1)(x-2)}$

Conclusion

And there you have it! We have successfully solved the rational expression $\frac{5}{x+1} - \frac{4}{x-2}$. By following the steps outlined in this article, you should be able to solve similar rational expressions with ease.

Final Answer

The final answer is $\boxed{\frac{x - 14}{(x+1)(x-2)}}$.

Additional Resources

If you are struggling with rational expressions or need additional practice, I recommend checking out the following resources:

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

FAQs

Q: What is a rational expression? A: A rational expression is a fraction that contains variables in the numerator or denominator.

Q: How do I find a common denominator? A: To find a common denominator, multiply both denominators by the same value.

Q: How do I simplify a rational expression? A: To simplify a rational expression, combine like terms and cancel out any common factors.

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Introduction

Rational expressions can be a challenging topic for many students. In this article, we will answer some of the most frequently asked questions about rational expressions. Whether you are a student, teacher, or simply looking to brush up on your math skills, this article is for you.

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables in the numerator or denominator. It is called "rational" because it is the ratio of two polynomials.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, combine like terms and cancel out any common factors. You can also use the following steps:

1. Find a common denominator.
2. Rewrite the expression with the common denominator.
3. Simplify the expression by combining like terms.
4. Cancel out any common factors.

Q: How do I find a common denominator?

A: To find a common denominator, multiply both denominators by the same value. For example, if you have the expression $\frac{5}{x+1} - \frac{4}{x-2}$, you can find a common denominator by multiplying both denominators by $(x+1)(x-2)$.

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 $\frac{3}{4}$. A rational expression, on the other hand, is a fraction that contains variables in the numerator or denominator.

Q: Can I add or subtract rational expressions?

A: Yes, you can add or subtract rational expressions. To do this, you need to find a common denominator and then combine the fractions.

Q: Can I multiply or divide rational expressions?

A: Yes, you can multiply or divide rational expressions. To do this, you can simply multiply or divide the numerators and denominators separately.

Q: What is the final answer to the expression $\frac{5}{x+1} - \frac{4}{x-2}$?

A: The final answer to the expression $\frac{5}{x+1} - \frac{4}{x-2}$ is $\boxed{\frac{x - 14}{(x+1)(x-2)}}$.

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

A: A rational expression is undefined if the denominator is equal to zero. For example, the expression $\frac{1}{x}$ is undefined when $x=0$.

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

A: Yes, you can simplify a rational expression with a variable in the denominator. To do this, you need to find a common denominator and then combine the fractions.

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

A: A rational expression is a fraction that contains variables in the numerator or denominator. An algebraic expression, on the other hand, is a general expression that can contain variables, constants, and mathematical operations.

Conclusion

We hope this article has helped you understand rational expressions better. Whether you are a student, teacher, or simply looking to brush up on your math skills, we hope this article has been helpful. If you have any further questions, please don't hesitate to ask.

Additional Resources

If you are struggling with rational expressions or need additional practice, I recommend checking out the following resources:

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

Final Answer

The final answer is $\boxed{\frac{x - 14}{(x+1)(x-2)}}$.