Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![The following mathematical expression needs to be simplified:
\[ \frac{5}{x^2 - 5x} - \frac{x}{5x - 25} \]
In this equation, two rational expressions are being subtracted from each other. Each rational expression consists of a numerator and a denominator:
1. The first fraction:
- Numerator: 5
- Denominator: \( x^2 - 5x \)
2. The second fraction:
- Numerator: x
- Denominator: \( 5x - 25 \)
To proceed with simplifying this expression, one may start by factoring the denominators.
### Factoring the Denominators
1. For the denominator \( x^2 - 5x \):
\[ x^2 - 5x = x(x - 5) \]
2. For the denominator \( 5x - 25 \):
\[ 5x - 25 = 5(x - 5) \]
### Simplified Expression
With these factorizations, the expression becomes:
\[ \frac{5}{x(x - 5)} - \frac{x}{5(x - 5)} \]
Combining the two fractions over a common denominator, which would be \( 5x(x - 5) \):
\[ \frac{5 \cdot 5 - x \cdot x}{5x(x - 5)} = \frac{25 - x^2}{5x(x - 5)} \]
### Final Notes
This simplified fraction can then be further analyzed or evaluated based on the specific values of \( x \), keeping in mind the restrictions where the denominator cannot be zero. In other words, \( x \neq 0 \) and \( x \neq 5 \) to avoid division by zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feefe174a-b7c5-4653-b479-736ad8d1161e%2F4ceddb5e-cbbe-4859-8217-dd5b26599ca9%2F06p5yvq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The following mathematical expression needs to be simplified:
\[ \frac{5}{x^2 - 5x} - \frac{x}{5x - 25} \]
In this equation, two rational expressions are being subtracted from each other. Each rational expression consists of a numerator and a denominator:
1. The first fraction:
- Numerator: 5
- Denominator: \( x^2 - 5x \)
2. The second fraction:
- Numerator: x
- Denominator: \( 5x - 25 \)
To proceed with simplifying this expression, one may start by factoring the denominators.
### Factoring the Denominators
1. For the denominator \( x^2 - 5x \):
\[ x^2 - 5x = x(x - 5) \]
2. For the denominator \( 5x - 25 \):
\[ 5x - 25 = 5(x - 5) \]
### Simplified Expression
With these factorizations, the expression becomes:
\[ \frac{5}{x(x - 5)} - \frac{x}{5(x - 5)} \]
Combining the two fractions over a common denominator, which would be \( 5x(x - 5) \):
\[ \frac{5 \cdot 5 - x \cdot x}{5x(x - 5)} = \frac{25 - x^2}{5x(x - 5)} \]
### Final Notes
This simplified fraction can then be further analyzed or evaluated based on the specific values of \( x \), keeping in mind the restrictions where the denominator cannot be zero. In other words, \( x \neq 0 \) and \( x \neq 5 \) to avoid division by zero.
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