Subtract The Following Expression:$\frac{4x+3}{x-5}-\frac{3x+3}{x-5}$

Introduction

When it comes to simplifying rational expressions, one of the most common operations is subtraction. In this article, we will focus on subtracting two rational expressions, specifically the expression $\frac{4x+3}{x-5}-\frac{3x+3}{x-5}$. We will break down the steps involved in subtracting these expressions and provide a clear, step-by-step guide on how to simplify the resulting expression.

Understanding Rational Expressions

Before we dive into the subtraction process, it's essential to understand what rational expressions are. A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by factoring the numerator and denominator, canceling out any common factors, and then simplifying the resulting expression.

The Expression to be Subtracted

The expression we will be working with is $\frac{4x+3}{x-5}-\frac{3x+3}{x-5}$. This expression consists of two rational expressions with the same denominator, $x-5$. Our goal is to subtract these two expressions and simplify the resulting expression.

Step 1: Factor the Numerators

To simplify the expression, we need to factor the numerators of both rational expressions. The first numerator, $4x+3$, can be factored as $(4x+3)$. The second numerator, $3x+3$, can be factored as $(3x+3)$.

Step 2: Rewrite the Expression with Factored Numerators

Now that we have factored the numerators, we can rewrite the expression with the factored numerators:

$$\frac{(4x+3)}{x-5}-\frac{(3x+3)}{x-5}$$

Step 3: Subtract the Numerators

Since the denominators are the same, we can subtract the numerators directly:

$$\frac{(4x+3)-(3x+3)}{x-5}$$

Step 4: Simplify the Numerator

Now, we need to simplify the numerator by combining like terms:

$$(4x+3)-(3x+3) = 4x-3x+3-3 = x$$

Step 5: Rewrite the Expression with the Simplified Numerator

Now that we have simplified the numerator, we can rewrite the expression with the simplified numerator:

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

Conclusion

In this article, we have walked through the steps involved in subtracting the rational expression $\frac{4x+3}{x-5}-\frac{3x+3}{x-5}$. We factored the numerators, rewrote the expression with factored numerators, subtracted the numerators, simplified the numerator, and finally rewrote the expression with the simplified numerator. The resulting expression is $\frac{x}{x-5}$.

Final Answer

The final answer is $\boxed{\frac{x}{x-5}}$.

Common Mistakes to Avoid

When subtracting rational expressions, it's essential to remember the following common mistakes to avoid:

• Not factoring the numerators
• Not rewriting the expression with factored numerators
• Not subtracting the numerators directly
• Not simplifying the numerator
• Not rewriting the expression with the simplified numerator

Real-World Applications

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

• Algebraic geometry
• Number theory
• Cryptography
• Electrical engineering
• Computer science

Future Directions

In the future, we can explore more advanced topics in rational expressions, such as:

• Adding and subtracting rational expressions with different denominators
• Multiplying and dividing rational expressions
• Simplifying rational expressions with complex denominators
• Using rational expressions in real-world applications

References

• [1] "Rational Expressions" by Math Open Reference
• [2] "Simplifying Rational Expressions" by Khan Academy
• [3] "Rational Expressions in Algebra" by Purplemath

Glossary

• Rational Expression: A fraction that contains variables and/or constants in the numerator and/or denominator.
• Denominator: The bottom part of a fraction.
• Numerator: The top part of a fraction.
• Like Terms: Terms that have the same variable and exponent.
• Simplify: To reduce an expression to its simplest form.

FAQs

• Q: What is a rational expression? A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.
• 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 resulting expression.
• Q: What is the difference between a rational expression and a polynomial? A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator, while a polynomial is an expression that consists of variables and/or constants multiplied together.
  

Introduction

Rational expressions are a fundamental concept in algebra, and understanding them is crucial for solving various mathematical problems. In this article, we will address some of the most frequently asked questions about rational expressions, providing clear and concise answers to help you better understand this topic.

Q&A

Q1: What is a rational expression?

A1: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q2: How do I simplify a rational expression?

A2: To simplify a rational expression, you need to factor the numerator and denominator, cancel out any common factors, and then simplify the resulting expression.

Q3: What is the difference between a rational expression and a polynomial?

A3: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator, while a polynomial is an expression that consists of variables and/or constants multiplied together.

Q4: How do I add and subtract rational expressions?

A4: To add and subtract rational expressions, you need to have the same denominator. If the denominators are different, you need to find the least common multiple (LCM) of the denominators and rewrite the expressions with the LCM as the denominator.

Q5: How do I multiply and divide rational expressions?

A5: To multiply rational expressions, you need to multiply the numerators and denominators separately. To divide rational expressions, you need to invert the second expression and multiply.

Q6: What is the least common multiple (LCM) of two expressions?

A6: The LCM of two expressions is the smallest expression that is a multiple of both expressions.

Q7: How do I simplify a rational expression with a complex denominator?

A7: To simplify a rational expression with a complex denominator, you need to factor the denominator and cancel out any common factors.

Q8: Can I simplify a rational expression with a variable in the denominator?

A8: Yes, you can simplify a rational expression with a variable in the denominator by factoring the denominator and canceling out any common factors.

Q9: How do I use rational expressions in real-world applications?

A9: Rational expressions are used in various real-world applications, such as algebraic geometry, number theory, cryptography, electrical engineering, and computer science.

Q10: What are some common mistakes to avoid when working with rational expressions?

A10: Some common mistakes to avoid when working with rational expressions include not factoring the numerator and denominator, not rewriting the expression with factored numerators, not subtracting the numerators directly, not simplifying the numerator, and not rewriting the expression with the simplified numerator.

Additional Resources

• [1] "Rational Expressions" by Math Open Reference
• [2] "Simplifying Rational Expressions" by Khan Academy
• [3] "Rational Expressions in Algebra" by Purplemath

Glossary

• Rational Expression: A fraction that contains variables and/or constants in the numerator and/or denominator.
• Denominator: The bottom part of a fraction.
• Numerator: The top part of a fraction.
• Like Terms: Terms that have the same variable and exponent.
• Simplify: To reduce an expression to its simplest form.
• Least Common Multiple (LCM): The smallest expression that is a multiple of two or more expressions.

FAQs

• Q: What is a rational expression? A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.
• 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 resulting expression.
• Q: What is the difference between a rational expression and a polynomial? A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator, while a polynomial is an expression that consists of variables and/or constants multiplied together.

Conclusion

Rational expressions are a fundamental concept in algebra, and understanding them is crucial for solving various mathematical problems. By following the steps outlined in this article, you can simplify rational expressions and apply them to real-world applications. Remember to avoid common mistakes and use additional resources to reinforce your understanding of rational expressions.