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
![Here is the problem to simplify:
\[
\frac{5x}{x^2 + 2x - 8} - \frac{3}{x + 4}
\]
In this problem, we are given two rational expressions to simplify by combining them.
First, notice the denominator in the first fraction, \(x^2 + 2x - 8\), can be factored. The factorization is:
\[
x^2 + 2x - 8 = (x + 4)(x - 2)
\]
Rewriting the expression using this factorization gives us:
\[
\frac{5x}{(x + 4)(x - 2)} - \frac{3}{x + 4}
\]
Next, we need a common denominator to combine these fractions. The common denominator will be \((x + 4)(x - 2)\).
Rewriting the second fraction with this common denominator, we have:
\[
\frac{3}{x + 4} = \frac{3(x - 2)}{(x + 4)(x - 2)} = \frac{3x - 6}{(x + 4)(x - 2)}
\]
Now, combining the fractions over the common denominator gives:
\[
\frac{5x - (3x - 6)}{(x + 4)(x - 2)} = \frac{5x - 3x + 6}{(x + 4)(x - 2)} = \frac{2x + 6}{(x + 4)(x - 2)}
\]
Hence, the simplified form of the given expression is:
\[
\frac{2x + 6}{(x + 4)(x - 2)}
\]
This completes the process of simplifying the given rational expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d158991-3671-4459-aef6-bd0615668acc%2F0439fb71-9004-44f6-9040-16590ba10761%2Fmin9cum_processed.png&w=3840&q=75)
Transcribed Image Text:Here is the problem to simplify:
\[
\frac{5x}{x^2 + 2x - 8} - \frac{3}{x + 4}
\]
In this problem, we are given two rational expressions to simplify by combining them.
First, notice the denominator in the first fraction, \(x^2 + 2x - 8\), can be factored. The factorization is:
\[
x^2 + 2x - 8 = (x + 4)(x - 2)
\]
Rewriting the expression using this factorization gives us:
\[
\frac{5x}{(x + 4)(x - 2)} - \frac{3}{x + 4}
\]
Next, we need a common denominator to combine these fractions. The common denominator will be \((x + 4)(x - 2)\).
Rewriting the second fraction with this common denominator, we have:
\[
\frac{3}{x + 4} = \frac{3(x - 2)}{(x + 4)(x - 2)} = \frac{3x - 6}{(x + 4)(x - 2)}
\]
Now, combining the fractions over the common denominator gives:
\[
\frac{5x - (3x - 6)}{(x + 4)(x - 2)} = \frac{5x - 3x + 6}{(x + 4)(x - 2)} = \frac{2x + 6}{(x + 4)(x - 2)}
\]
Hence, the simplified form of the given expression is:
\[
\frac{2x + 6}{(x + 4)(x - 2)}
\]
This completes the process of simplifying the given rational expressions.
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