Question

\\\frac{3}{x - 3} + \frac{x}{x + 3} =\\

(a) \\\frac{1}{x - 3}\\
(b) \\\frac{x + 3}{x - 3}\\
(c) \\\frac{3 + x}{2x}\\
(d) \\\frac{x^2}{x^2 - 9}\\
(e) \\\frac{x^2 + 9}{x^2 - 9}\\

Explanation:

<pre_analysis>
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"Rational Expressions",
"Common Denominator"
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"current_concepts": [
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}
</pre_analysis>

<reasoning>

Find a common denominator

\[
\text{LCD} = (x - 3)(x + 3) = x^2 - 9
\]

Rewrite each fraction with the common denominator

\[
\frac{3(x + 3)}{(x - 3)(x + 3)} + \frac{x(x - 3)}{(x - 3)(x + 3)}
\]

Combine and simplify the numerators

\[
\frac{3x + 9 + x^2 - 3x}{x^2 - 9} = \frac{x^2 + 9}{x^2 - 9}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{1}{x - 3}\)</mcq-option>
<mcq-option>(B) \(\frac{x + 3}{x - 3}\)</mcq-option>
<mcq-option>(C) \(\frac{3 + x}{2x}\)</mcq-option>
<mcq-option>(D) \(\frac{x^2}{x^2 - 9}\)</mcq-option>
<mcq-correct>(E) \(\frac{x^2 + 9}{x^2 - 9}\)</mcq-correct>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Rational Expressions"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
"multiple_choice_formatter",
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}
</pre_analysis>

<reasoning>

Find a common denominator

\[
\text{LCD} = (x - 3)(x + 3) = x^2 - 9
\]

Rewrite each fraction with the common denominator

\[
\frac{3(x + 3)}{(x - 3)(x + 3)} + \frac{x(x - 3)}{(x - 3)(x + 3)}
\]

Combine and simplify the numerators

\[
\frac{3x + 9 + x^2 - 3x}{x^2 - 9} = \frac{x^2 + 9}{x^2 - 9}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{1}{x - 3}\)</mcq-option>
<mcq-option>(B) \(\frac{x + 3}{x - 3}\)</mcq-option>
<mcq-option>(C) \(\frac{3 + x}{2x}\)</mcq-option>
<mcq-option>(D) \(\frac{x^2}{x^2 - 9}\)</mcq-option>
<mcq-correct>(E) \(\frac{x^2 + 9}{x^2 - 9}\)</mcq-correct>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Rational Expressions"
]
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