To find the sum of the rational expressions \( \frac{5x + 2}{1} + \frac{6x - 3}{1} \), we will first rewrite them with a common denominator.
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Combine the Numerators: \[ (5x + 2) + (6x - 3) = 5x + 2 + 6x - 3 = 11x - 1 \]
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Rewrite with a Denominator: Now we want to represent this as a single fraction. Since both expressions are over the same denominator (which we can use as 1): \[ \frac{11x - 1}{1} = 11x - 1 \]
However, to adjust this to the options given, let's consider the need for polynomial factors in the denominators, specifically \( x^2 - x - 6 \).
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Factor the Denominator: The expression \( x^2 - x - 6 \) factors to \( (x - 3)(x + 2) \).
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Attempt to Rewrite the Expression: Since \( x^2 - x - 6 \) isn't immediately a common denominator for this sum, we should construct the common denominator: \[ \frac{(11x - 1)\cdot{1}}{1} = \frac{11x - 1}{1} \cdot \frac{x^2 - x - 6}{x^2 - x - 6} = \frac{(11x - 1)(x^2 - x - 6)}{x^2 - x - 6} \]
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Adding Rational Expressions: But to form \( \frac{11x + C}{x^2 - x - 6} \), we may need actual individual rational forms that add up to a common base. In fact from our previous operations, we see: The complete fraction would remain like \( \frac{11x - 1}{1} \).
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Select from the Choices: The description initially lists:
- \( \frac{11x - 3}{x^2 - x - 6} \)
- \( \frac{11x + 27}{x^2 - x - 6} \)
- \( \frac{11}{x - 1} - 3x - 6 \)
Final Rational Expression: So through inspections: The first one captures \( 11x - 3 \) over \( x^2 - x - 6 \), though not quite matching \( -1 \).
Thus we will arrive at the closest value: The result greatly prefers or implies the option: Response: \(\frac{11x - 3}{x^2 - x - 6}\).
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