Which Is The Simplified Rational Expression For 9 X − 5 4 X − 1 + ( 3 X − 8 4 X − 1 − 5 X + 3 4 X − 1 ) ? \frac{9x-5}{4x-1} + \left(\frac{3x-8}{4x-1} - \frac{5x+3}{4x-1}\right)? 4 X − 1 9 X − 5 ​ + ( 4 X − 1 3 X − 8 ​ − 4 X − 1 5 X + 3 ​ ) ? A. 7 X − 10 4 X − 1 \frac{7x-10}{4x-1} 4 X − 1 7 X − 10 ​ B. 7 X − 16 4 X − 1 \frac{7x-16}{4x-1} 4 X − 1 7 X − 16 ​ C. 17 X − 10 4 X − 1 \frac{17x-10}{4x-1} 4 X − 1 17 X − 10 ​ D.

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Introduction

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Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore the process of simplifying rational expressions, with a focus on the given problem: $\frac{9x-5}{4x-1} + \left(\frac{3x-8}{4x-1} - \frac{5x+3}{4x-1}\right)$. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding Rational Expressions

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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 simplifying rational expressions, we need to be mindful of the rules of arithmetic operations.

Key Concepts

• Like Terms: Terms that have the same variable and exponent are called like terms. For example, $2x$ and $3x$ are like terms.
• Common Factors: Factors that are common to both the numerator and denominator are called common factors. For example, in the expression $\frac{6x}{3x}$, the common factor is $3x$.
• Simplifying Rational Expressions: To simplify a rational expression, we need to cancel out any common factors between the numerator and denominator.

Step 1: Simplify the Expression Inside the Parentheses

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The given expression is $\frac{9x-5}{4x-1} + \left(\frac{3x-8}{4x-1} - \frac{5x+3}{4x-1}\right)$. To simplify this expression, we need to start by simplifying the expression inside the parentheses.

Simplifying the Expression Inside the Parentheses

$\frac{3x-8}{4x-1} - \frac{5x+3}{4x-1}$

To simplify this expression, we need to find a common denominator, which is $4x-1$. We can then rewrite each fraction with the common denominator:

$\frac{(3x-8)(4x-1)}{(4x-1)(4x-1)} - \frac{(5x+3)(4x-1)}{(4x-1)(4x-1)}$

Simplifying the numerators, we get:

$\frac{12x^2-35x+8}{(4x-1)^2} - \frac{20x^2-17x-3}{(4x-1)^2}$

Now, we can combine the two fractions by adding or subtracting the numerators:

$\frac{(12x^2-35x+8) - (20x^2-17x-3)}{(4x-1)^2}$

Simplifying the numerator, we get:

$\frac{-8x^2+12x+11}{(4x-1)^2}$

Step 2: Simplify the Original Expression

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Now that we have simplified the expression inside the parentheses, we can simplify the original expression:

$\frac{9x-5}{4x-1} + \left(\frac{-8x^2+12x+11}{(4x-1)^2}\right)$

To simplify this expression, we need to find a common denominator, which is $(4x-1)^2$. We can then rewrite each fraction with the common denominator:

$\frac{(9x-5)(4x-1)}{(4x-1)^2} + \frac{(-8x^2+12x+11)(4x-1)}{(4x-1)^2}$

Simplifying the numerators, we get:

$\frac{36x^2-49x-5}{(4x-1)^2} + \frac{-32x^3+52x^2+44x-11}{(4x-1)^2}$

Now, we can combine the two fractions by adding or subtracting the numerators:

$\frac{(36x^2-49x-5) + (-32x^3+52x^2+44x-11)}{(4x-1)^2}$

Simplifying the numerator, we get:

$\frac{-32x^3+88x^2-5x-16}{(4x-1)^2}$

Step 3: Factor the Numerator

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The numerator of the simplified expression is $-32x^3+88x^2-5x-16$. We can factor this expression by grouping:

$-32x^3+88x^2-5x-16$

Factoring out $-1$, we get:

$-1(32x^3-88x^2+5x+16)$

Factoring out $4x$, we get:

$-4x(8x^2-22x-4)$

Factoring the quadratic expression, we get:

$-4x(4x-5)(2x+1)$

Step 4: Simplify the Expression

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Now that we have factored the numerator, we can simplify the expression:

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

To simplify this expression, we need to cancel out any common factors between the numerator and denominator. The common factor is $4x-1$, so we can cancel it out:

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

Simplifying the expression, we get:

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

Conclusion

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In this article, we have simplified the rational expression $\frac{9x-5}{4x-1} + \left(\frac{3x-8}{4x-1} - \frac{5x+3}{4x-1}\right)$. We have broken down the solution into manageable steps, making it easy to follow and understand. The final simplified expression is $\frac{-4x(4x-5)(2x+1)}{4x^2-2x-1}$.

Final Answer

The final answer is $\boxed{\frac{-4x(4x-5)(2x+1)}{4x^2-2x-1}}$.

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Introduction

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In our previous article, we explored the process of simplifying rational expressions, with a focus on the given problem: $\frac{9x-5}{4x-1} + \left(\frac{3x-8}{4x-1} - \frac{5x+3}{4x-1}\right)$. We broke down the solution into manageable steps, making it easy to follow and understand. In this article, we will provide a Q&A guide to help you better understand the concept of simplifying rational expressions.

Q&A Guide

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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 cancel out any common factors between the numerator and denominator.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $2x$ and $3x$ are like terms.

Q: What are common factors?

A: Common factors are factors that are common to both the numerator and denominator. For example, in the expression $\frac{6x}{3x}$, the common factor is $3x$.

Q: How do I find a common denominator?

A: To find a common denominator, you need to multiply the denominators of the two fractions together.

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

A: When adding rational expressions, you need to find a common denominator and then add the numerators. When subtracting rational expressions, you need to find a common denominator and then subtract the numerators.

Q: Can I simplify a rational expression by canceling out a common factor in the numerator and denominator?

A: Yes, you can simplify a rational expression by canceling out a common factor in the numerator and denominator.

Q: What is the final simplified expression for the given problem?

A: The final simplified expression for the given problem is $\frac{-4x(4x-5)(2x+1)}{4x^2-2x-1}$.

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

A: A rational expression is simplified when there are no common factors between the numerator and denominator.

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.

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

A: Some common mistakes to avoid when simplifying rational expressions include:

• Not canceling out common factors between the numerator and denominator
• Not finding a common denominator when adding or subtracting rational expressions
• Not simplifying the numerator and denominator separately

Conclusion

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In this article, we have provided a Q&A guide to help you better understand the concept of simplifying rational expressions. We have covered topics such as like terms, common factors, finding a common denominator, and simplifying rational expressions. We hope that this guide has been helpful in clarifying any confusion you may have had about simplifying rational expressions.

Final Tips

• Always simplify the numerator and denominator separately
• Cancel out common factors between the numerator and denominator
• Find a common denominator when adding or subtracting rational expressions
• Be careful when simplifying rational expressions with variables in the denominator

Final Answer

The final answer is $\boxed{\frac{-4x(4x-5)(2x+1)}{4x^2-2x-1}}$.