To determine which additional operation should be included in the expression \( \frac{5}{6}x + 6 + _ \) in order to make it equivalent to \( \frac{1}{2}x + 6 \), let's analyze the equation.
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Start with the expression we want: \[ \frac{1}{2}x + 6 \]
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The expression we currently have is: \[ \frac{5}{6}x + 6 + _ \]
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Since both expressions contain the term \( +6 \), we can focus on the \( x \) coefficients.
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The coefficient of \( x \) in the expression \( \frac{1}{2}x + 6 \) is \( \frac{1}{2} \).
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The coefficient of \( x \) in the expression \( \frac{5}{6}x + 6 \) is \( \frac{5}{6} \).
Now, we need to find a value that, when added to \( \frac{5}{6} \), will give us \( \frac{1}{2} \).
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We can set up the equation: \[ \frac{5}{6} + c = \frac{1}{2} \] where \( c \) is the additional amount we will add.
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Convert \( \frac{1}{2} \) to sixths so we can easily solve: \[ \frac{1}{2} = \frac{3}{6} \]
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Now we have: \[ \frac{5}{6} + c = \frac{3}{6} \]
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Rearranging gives: \[ c = \frac{3}{6} - \frac{5}{6} = -\frac{2}{6} = -\frac{1}{3} \]
Now, we have found that \( c \) (the additional term) must be \( -\frac{1}{3} \).
However, among the choices given:
- −3/4x
- −x
- −2x
- 3/4x
None of these choices equal \( -\frac{1}{3}x \).
Let's analyze each choice in terms of the impact on the coefficient of \( x \) to see which one could potentially result in a net equivalent coefficient of \( \frac{1}{2} \):
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Adding −3/4x would result in: \[ \frac{5}{6}x - \frac{3}{4}x \implies \left( \frac{5}{6} - \frac{9}{12} \right) = \left( \frac{10}{12} - \frac{9}{12} \right) = \frac{1}{12}x \quad (\text{not } \frac{1}{2}) \]
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Adding −x results in: \[ \frac{5}{6}x - 1x = \left( \frac{5}{6} - \frac{6}{6} \right) = -\frac{1}{6}x \quad (\text{not } \frac{1}{2}) \]
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Adding −2x results in: \[ \frac{5}{6}x - 2x = \left( \frac{5}{6} - \frac{12}{6} \right) = -\frac{7}{6}x \quad (\text{not } \frac{1}{2}) \]
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Adding 3/4x results in: \[ \frac{5}{6}x + \frac{3}{4}x = \left( \frac{10}{12} + \frac{9}{12} \right) = \frac{19}{12}x \quad (\text{not } \frac{1}{2}) \]
Since none of these options achieve the desired \( \frac{1}{2}x \), it appears they are incorrect in providing equivalency as expected.
Based on the coefficients given to the operation of \(-\frac{1}{3}\) for \( x \), plus the terms given, it seems there is a mismatch or misunderstanding in potential operation terms you've shared.
The correct answer must be \( -\frac{1}{3} \) if given as a choice for the equation to balance to \( \frac{1}{2}x \) but in provided choices, there isn't a viable match or resolution towards attaining desired values from existing options.