Simplify The Expression:$\[ \frac{5}{x+3} - \frac{3}{2-x} \\]

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and mathematician should possess. In this article, we will delve into the world of algebraic manipulation and provide a step-by-step guide on how to simplify the given expression: $\frac{5}{x+3} - \frac{3}{2-x}$. We will explore various techniques and strategies to simplify the expression, making it easier to understand and work with.

Understanding the Expression

The given expression is a combination of two fractions: $\frac{5}{x+3}$ and $\frac{3}{2-x}$. To simplify the expression, we need to first understand the properties of fractions and how they can be manipulated. A fraction is a way of representing a part of a whole, and it consists of a numerator and a denominator. In this case, the numerator is the number on top, and the denominator is the number on the bottom.

Simplifying the Expression

To simplify the expression, we need to find a common denominator for both fractions. The common denominator is the least common multiple (LCM) of the two denominators. In this case, the LCM of $(x+3)$ and $(2-x)$ is $(x+3)(2-x)$.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the expression
expr = 5/(x+3) - 3/(2-x)

# Simplify the expression
simplified_expr = sp.simplify(expr)

print(simplified_expr)

Using the Least Common Multiple (LCM)

The LCM of $(x+3)$ and $(2-x)$ is $(x+3)(2-x)$. To find the LCM, we can use the following steps:

1. Factorize both denominators: $(x+3)$ and $(2-x)$.
2. Identify the common factors: $1$.
3. Multiply the common factors: $1 \times 1 = 1$.
4. Multiply the remaining factors: $(x+3) \times (2-x) = 2x + 6 - x - 3 = x + 3$.

Combining the Fractions

Now that we have found the LCM, we can combine the fractions by multiplying the numerator and denominator of each fraction by the LCM.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the expression
expr = 5/(x+3) - 3/(2-x)

# Combine the fractions
combined_expr = (5*(2-x) - 3*(x+3))/((x+3)*(2-x))

print(combined_expr)

Simplifying the Numerator

Now that we have combined the fractions, we can simplify the numerator by multiplying the terms.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the expression
expr = (5*(2-x) - 3*(x+3))/((x+3)*(2-x))

# Simplify the numerator
simplified_numerator = 10 - 5*x - 9 - 3*x

print(simplified_numerator)

Final Simplification

Now that we have simplified the numerator, we can simplify the entire expression by combining the terms.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the expression
expr = (10 - 5*x - 9 - 3*x)/((x+3)*(2-x))

# Simplify the expression
simplified_expr = sp.simplify(expr)

print(simplified_expr)

Conclusion

In this article, we have provided a step-by-step guide on how to simplify the given expression: $\frac{5}{x+3} - \frac{3}{2-x}$. We have used various techniques and strategies to simplify the expression, including finding the least common multiple (LCM) and combining the fractions. By following these steps, we have arrived at the final simplified expression.

Final Answer

The final simplified expression is: $\boxed{\frac{-7x+1}{(x+3)(2-x)}}$

Discussion

The expression $\frac{5}{x+3} - \frac{3}{2-x}$ can be simplified using various techniques and strategies. In this article, we have used the least common multiple (LCM) and combining the fractions to simplify the expression. However, there are other ways to simplify the expression, and we encourage readers to explore these alternative methods.

Additional Resources

For further reading and practice, we recommend the following resources:

• Khan Academy: Algebra
• MIT OpenCourseWare: Algebra
• Wolfram Alpha: Algebra

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Note: The references provided are for further reading and practice, and are not directly related to the content of this article.

Introduction

In our previous article, we provided a step-by-step guide on how to simplify the given expression: $\frac{5}{x+3} - \frac{3}{2-x}$. We explored various techniques and strategies to simplify the expression, including finding the least common multiple (LCM) and combining the fractions. In this article, we will answer some of the most frequently asked questions (FAQs) related to simplifying the expression.

Q&A

Q1: What is the least common multiple (LCM) of $(x+3)$ and $(2-x)$?

A1: The LCM of $(x+3)$ and $(2-x)$ is $(x+3)(2-x)$.

Q2: How do I find the LCM of two expressions?

A2: To find the LCM of two expressions, you can use the following steps:

1. Factorize both expressions.
2. Identify the common factors.
3. Multiply the common factors.
4. Multiply the remaining factors.

Q3: What is the difference between the numerator and denominator of the combined fraction?

A3: The numerator of the combined fraction is $5(2-x) - 3(x+3)$, and the denominator is $(x+3)(2-x)$.

Q4: How do I simplify the numerator of the combined fraction?

A4: To simplify the numerator, you can multiply the terms and combine like terms.

Q5: What is the final simplified expression?

A5: The final simplified expression is: $\boxed{\frac{-7x+1}{(x+3)(2-x)}}$

Q6: Can I use other techniques to simplify the expression?

A6: Yes, there are other techniques and strategies that you can use to simplify the expression. We encourage readers to explore these alternative methods.

Q7: What are some common mistakes to avoid when simplifying expressions?

A7: Some common mistakes to avoid when simplifying expressions include:

• Not finding the least common multiple (LCM) of the denominators.
• Not combining the fractions correctly.
• Not simplifying the numerator and denominator separately.

Q8: How can I practice simplifying expressions?

A8: You can practice simplifying expressions by working through examples and exercises. We recommend using online resources, such as Khan Academy and Wolfram Alpha, to find practice problems and exercises.

Conclusion

In this article, we have answered some of the most frequently asked questions (FAQs) related to simplifying the expression: $\frac{5}{x+3} - \frac{3}{2-x}$. We have provided step-by-step guides and explanations to help readers understand the techniques and strategies used to simplify the expression. We encourage readers to practice simplifying expressions and to explore alternative methods.

Final Answer

The final simplified expression is: $\boxed{\frac{-7x+1}{(x+3)(2-x)}}$

Discussion

The expression $\frac{5}{x+3} - \frac{3}{2-x}$ can be simplified using various techniques and strategies. In this article, we have used the least common multiple (LCM) and combining the fractions to simplify the expression. However, there are other ways to simplify the expression, and we encourage readers to explore these alternative methods.

Additional Resources

For further reading and practice, we recommend the following resources:

• Khan Academy: Algebra
• MIT OpenCourseWare: Algebra
• Wolfram Alpha: Algebra

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Note: The references provided are for further reading and practice, and are not directly related to the content of this article.