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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![### Example Problem: Simplifying Rational Expressions
#### Problem Statement:
Simplify the following expression:
\[
\frac{x + 4}{4x + 24} - \frac{3}{x^2 + 6x}
\]
#### Steps for Simplification:
1. **Factor the Denominators:**
\(4x + 24\) and \(x^2 + 6x\) need to be factored to find a common denominator.
- \(4x + 24\) can be factored as \(4(x + 6)\).
- \(x^2 + 6x\) can be factored as \(x(x + 6)\).
So, the expression becomes:
\[
\frac{x + 4}{4(x + 6)} - \frac{3}{x(x + 6)}
\]
2. **Find a Common Denominator:**
The least common denominator (LCD) for \(4(x + 6)\) and \(x(x + 6)\) is \(4x(x + 6)\).
3. **Write Each Fraction with the Common Denominator:**
Convert each term of the expression to have the common denominator \(4x(x + 6)\):
- \(\frac{x + 4}{4(x + 6)}\) becomes \(\frac{x + 4}{4(x + 6)} \cdot \frac{x}{x} = \frac{x(x + 4)}{4x(x + 6)}\)
- \(\frac{3}{x(x + 6)}\) becomes \(\frac{3}{x(x + 6)} \cdot \frac{4}{4} = \frac{12}{4x(x + 6)}\)
Now, the expression is:
\[
\frac{x(x + 4)}{4x(x + 6)} - \frac{12}{4x(x + 6)}
\]
4. **Combine the Numerators:**
Combine the numerators over the common denominator:
\[
\frac{x(x + 4) - 12}{4x(x + 6)}
\]
5. **Simplify the Numerator:**
Simplify \( x(x + 4) - 12 \):
- Expand \(x(x +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d158991-3671-4459-aef6-bd0615668acc%2F1c8fc699-a8aa-4939-8c3e-85a8bf3ccc8d%2Fxb22jrb_processed.png&w=3840&q=75)
Transcribed Image Text:### Example Problem: Simplifying Rational Expressions
#### Problem Statement:
Simplify the following expression:
\[
\frac{x + 4}{4x + 24} - \frac{3}{x^2 + 6x}
\]
#### Steps for Simplification:
1. **Factor the Denominators:**
\(4x + 24\) and \(x^2 + 6x\) need to be factored to find a common denominator.
- \(4x + 24\) can be factored as \(4(x + 6)\).
- \(x^2 + 6x\) can be factored as \(x(x + 6)\).
So, the expression becomes:
\[
\frac{x + 4}{4(x + 6)} - \frac{3}{x(x + 6)}
\]
2. **Find a Common Denominator:**
The least common denominator (LCD) for \(4(x + 6)\) and \(x(x + 6)\) is \(4x(x + 6)\).
3. **Write Each Fraction with the Common Denominator:**
Convert each term of the expression to have the common denominator \(4x(x + 6)\):
- \(\frac{x + 4}{4(x + 6)}\) becomes \(\frac{x + 4}{4(x + 6)} \cdot \frac{x}{x} = \frac{x(x + 4)}{4x(x + 6)}\)
- \(\frac{3}{x(x + 6)}\) becomes \(\frac{3}{x(x + 6)} \cdot \frac{4}{4} = \frac{12}{4x(x + 6)}\)
Now, the expression is:
\[
\frac{x(x + 4)}{4x(x + 6)} - \frac{12}{4x(x + 6)}
\]
4. **Combine the Numerators:**
Combine the numerators over the common denominator:
\[
\frac{x(x + 4) - 12}{4x(x + 6)}
\]
5. **Simplify the Numerator:**
Simplify \( x(x + 4) - 12 \):
- Expand \(x(x +
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