Elementary Algebra
17th Edition
ISBN:9780998625713
Author:Lynn Marecek, MaryAnne Anthony-Smith
Publisher:Lynn Marecek, MaryAnne Anthony-Smith
Chapter8: Rational Expressions And Equations
Section8.3: Add And Subtract Rational Expressions With A Common Denominator
Problem 166E: If Pete can paint a wall in p hours, then in one hour he can paint 1p of the wall. It would take...
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Question
Perform the operation
![**Problem Statement:**
Simplify the following rational expression:
\[ \frac{3x - 2}{9x - 1} + \frac{5x - 4}{x - 3} \]
This problem requires the addition of two rational expressions. To add these fractions, a common denominator should be found and used to combine the two fractions into a single expression.
**Detailed Steps to Solve:**
1. **Identifying the Denominators:**
The given rational expressions are \(\frac{3x - 2}{9x - 1}\) and \(\frac{5x - 4}{x - 3}\). The denominators are \(9x - 1\) and \(x - 3\).
2. **Finding the Least Common Denominator (LCD):**
The least common denominator of \(9x - 1\) and \(x - 3\) is obtained by multiplying these distinct polynomial expressions together, giving us \((9x - 1)(x - 3)\).
3. **Rewriting Each Fraction with the LCD:**
Each fraction is then rewritten with the LCD:
\[
\frac{3x - 2}{9x - 1} = \frac{(3x - 2)(x - 3)}{(9x - 1)(x - 3)}
\]
\[
\frac{5x - 4}{x - 3} = \frac{(5x - 4)(9x - 1)}{(9x - 1)(x - 3)}
\]
4. **Combining the Fractions:**
Adding these fractions:
\[
\frac{(3x - 2)(x - 3) + (5x - 4)(9x - 1)}{(9x - 1)(x - 3)}
\]
5. **Simplifying the Numerator:**
Expand and combine like terms in the numerator:
\[
(3x - 2)(x - 3) + (5x - 4)(9x - 1)
\]
6. **Final Expression:**
Once the numerator is simplified, you will have the final rational expression.
**Further Explanation:**
This process may involve multiple steps of algebraic expansion and simplification](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa38d3af-d2a1-4c36-a58b-a982b5534c39%2F58f51a85-bc14-47cc-babf-8b49c0a02d79%2Fnk27wtm.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Simplify the following rational expression:
\[ \frac{3x - 2}{9x - 1} + \frac{5x - 4}{x - 3} \]
This problem requires the addition of two rational expressions. To add these fractions, a common denominator should be found and used to combine the two fractions into a single expression.
**Detailed Steps to Solve:**
1. **Identifying the Denominators:**
The given rational expressions are \(\frac{3x - 2}{9x - 1}\) and \(\frac{5x - 4}{x - 3}\). The denominators are \(9x - 1\) and \(x - 3\).
2. **Finding the Least Common Denominator (LCD):**
The least common denominator of \(9x - 1\) and \(x - 3\) is obtained by multiplying these distinct polynomial expressions together, giving us \((9x - 1)(x - 3)\).
3. **Rewriting Each Fraction with the LCD:**
Each fraction is then rewritten with the LCD:
\[
\frac{3x - 2}{9x - 1} = \frac{(3x - 2)(x - 3)}{(9x - 1)(x - 3)}
\]
\[
\frac{5x - 4}{x - 3} = \frac{(5x - 4)(9x - 1)}{(9x - 1)(x - 3)}
\]
4. **Combining the Fractions:**
Adding these fractions:
\[
\frac{(3x - 2)(x - 3) + (5x - 4)(9x - 1)}{(9x - 1)(x - 3)}
\]
5. **Simplifying the Numerator:**
Expand and combine like terms in the numerator:
\[
(3x - 2)(x - 3) + (5x - 4)(9x - 1)
\]
6. **Final Expression:**
Once the numerator is simplified, you will have the final rational expression.
**Further Explanation:**
This process may involve multiple steps of algebraic expansion and simplification
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