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Chapter 7: Problem 63

Perform the indicated operation. Simplify, if possible. $$ \frac{3 x+2}{3 x+6}+\frac{x}{4-x^{2}} $$

Short Answer

Expert verified
\[ \frac{4 + 3x - x^2}{3(2 + x)(2 - x)} \]

Step by step solution

01

Factor the denominators

Identify any factors in the denominators that can be factored. The first denominator, \( 3x + 6 \), can be factored as \( 3(x + 2) \). The second denominator, \( 4 - x^2 \), is a difference of squares and can be factored as \( (2 + x)(2 - x) \). So the expression becomes: \[ \frac{3x + 2}{3(x + 2)} + \frac{x}{(2 + x)(2 - x)} \]
02

Simplify the fractions

Cancel out the common factors in the numerators and denominators of each fraction. For the first fraction, \( 3x + 2 \) and \( x + 2 \) cancel out partially, simplifying to \[ \frac{1}{3} \] For the second fraction, \( x \) doesn't cancel, so it remains as \[ \frac{x}{(2 + x)(2 - x)} \]
03

Find the common denominator

To add the fractions, we'll need a common denominator. The denominators are \( 3 \) and \((2 + x)(2 - x) \). Therefore, the common denominator will be \3(2 + x)(2 - x)\.
04

Rewrite each fraction with the common denominator

Convert each fraction so they have the common denominator \3(2 + x)(2 - x)\. The first fraction becomes: \[ \frac{1}{3} = \frac{(2 + x)(2 - x)}{3(2 + x)(2 - x)} = \frac{4 - x^2}{3(2 + x)(2 - x)} \] \ The second fraction remains: \[ \frac{x}{(2 + x)(2 - x)} = \frac{3x}{3(2 + x)(2 - x)} \]
05

Add the fractions

Add the two fractions together: \[ \frac{4 - x^2 + 3x}{3(2 + x)(2 - x)} \]
06

Simplify the numerator

Combine like terms in the numerator: \[ \frac{4 - x^2 + 3x}{3(2 + x)(2 - x)} = \frac{4 + 3x - x^2}{3(2 + x)(2 - x)} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
The final step after adding fractions is to simplify the resulting expression.
This involves combining like terms in the numerator to make the expression as simple as possible.
In our case, we add the numerators: \[ 4 - x^2 + 3x \] and write it over the common denominator: \[ \frac{4 + 3x - x^2}{3(2 + x)(2 - x)} \]
This is the simplest form of our expression after performing the operation and simplifying.

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