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
![### Combining Fractions into a Single Fraction
To combine these two fractions into a single fraction, follow these steps:
#### Problem:
\[
\frac{4}{x-2} - \frac{3}{x}
\]
#### Step-by-Step Solution:
1. **Find a common denominator**:
The least common denominator (LCD) of \(x - 2\) and \(x\) is \((x - 2) \cdot x\).
2. **Rewrite each fraction with the common denominator**:
\[
\frac{4}{x-2} = \frac{4x}{x(x-2)}
\]
\[
\frac{3}{x} = \frac{3(x-2)}{x(x-2)}
\]
3. **Express each fraction using the common denominator**:
\[
\frac{4x}{x(x-2)} - \frac{3(x-2)}{x(x-2)}
\]
4. **Combine the numerators over the same denominator**:
\[
\frac{4x - 3(x-2)}{x(x-2)}
\]
5. **Simplify the numerator**:
\[
4x - 3(x-2) = 4x - 3x + 6 = x + 6
\]
Therefore,
\[
\frac{x + 6}{x(x-2)}
\]
#### Final Answer:
\[
\frac{x + 6}{x(x-2)}
\]
This is the single fraction that combines the original two fractions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c9551-2207-4a1f-96cc-82c211723c9b%2F4602a94f-01e2-4126-b0e3-f009c2462205%2Fbk7mubj_processed.png&w=3840&q=75)
Transcribed Image Text:### Combining Fractions into a Single Fraction
To combine these two fractions into a single fraction, follow these steps:
#### Problem:
\[
\frac{4}{x-2} - \frac{3}{x}
\]
#### Step-by-Step Solution:
1. **Find a common denominator**:
The least common denominator (LCD) of \(x - 2\) and \(x\) is \((x - 2) \cdot x\).
2. **Rewrite each fraction with the common denominator**:
\[
\frac{4}{x-2} = \frac{4x}{x(x-2)}
\]
\[
\frac{3}{x} = \frac{3(x-2)}{x(x-2)}
\]
3. **Express each fraction using the common denominator**:
\[
\frac{4x}{x(x-2)} - \frac{3(x-2)}{x(x-2)}
\]
4. **Combine the numerators over the same denominator**:
\[
\frac{4x - 3(x-2)}{x(x-2)}
\]
5. **Simplify the numerator**:
\[
4x - 3(x-2) = 4x - 3x + 6 = x + 6
\]
Therefore,
\[
\frac{x + 6}{x(x-2)}
\]
#### Final Answer:
\[
\frac{x + 6}{x(x-2)}
\]
This is the single fraction that combines the original two fractions.
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