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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Question
![### Algebraic Equation Example
In this example, we are working with a rational equation that involves polynomials in the numerator and the denominator. The equation is given as:
\[ \frac{5x + 5}{(x + 3)} + \frac{x - 1}{(x - 4)} = \frac{6x}{(x + 3)} \]
This equation can be broken down as follows:
1. **First Term**: \[ \frac{5x + 5}{(x + 3)} \]
2. **Second Term**: \[ \frac{x - 1}{(x - 4)} \]
3. **Right-Hand Side**: \[ \frac{6x}{(x + 3)} \]
### Solving the Equation
To solve this equation, one common method is to find a common denominator and then equate the numerators, simplifying as necessary. Here are the steps involved:
1. **Identify the common denominator**, which in this case would be \((x + 3)(x - 4)\).
2. **Rewrite each term with the common denominator**.
3. **Combine the terms** on the left-hand side of the equation.
4. **Equate the numerators** on both sides of the equation.
5. **Solve the resulting polynomial equation**.
This process involves several algebraic manipulation techniques, including polynomial multiplication, combining like terms, and factoring.
This example is intended to highlight the method for solving rational equations with polynomial expressions. Working through examples like this can help solidify understanding of these algebraic concepts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc1123ec-a0e0-4e35-afe0-df78f47a58b5%2Fd3914385-1f05-4864-a539-96dad1030439%2Fm0bme4i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Algebraic Equation Example
In this example, we are working with a rational equation that involves polynomials in the numerator and the denominator. The equation is given as:
\[ \frac{5x + 5}{(x + 3)} + \frac{x - 1}{(x - 4)} = \frac{6x}{(x + 3)} \]
This equation can be broken down as follows:
1. **First Term**: \[ \frac{5x + 5}{(x + 3)} \]
2. **Second Term**: \[ \frac{x - 1}{(x - 4)} \]
3. **Right-Hand Side**: \[ \frac{6x}{(x + 3)} \]
### Solving the Equation
To solve this equation, one common method is to find a common denominator and then equate the numerators, simplifying as necessary. Here are the steps involved:
1. **Identify the common denominator**, which in this case would be \((x + 3)(x - 4)\).
2. **Rewrite each term with the common denominator**.
3. **Combine the terms** on the left-hand side of the equation.
4. **Equate the numerators** on both sides of the equation.
5. **Solve the resulting polynomial equation**.
This process involves several algebraic manipulation techniques, including polynomial multiplication, combining like terms, and factoring.
This example is intended to highlight the method for solving rational equations with polynomial expressions. Working through examples like this can help solidify understanding of these algebraic concepts.
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