To determine which additional operation should be included in the expression \( \frac{5}{4}x + 6 + _ \) so that it equals \( \frac{1}{2}x + 6 \), we can set up the equation:
\[ \frac{5}{4}x + 6 + _ = \frac{1}{2}x + 6 \]
Subtracting \( 6 \) from both sides gives us:
\[ \frac{5}{4}x + _ = \frac{1}{2}x \]
Next, we need to isolate the additional operation \( _ \) by rearranging the equation:
\[ _ = \frac{1}{2}x - \frac{5}{4}x \]
To perform this subtraction, we need a common denominator, which will be \( 4 \):
\[ \frac{1}{2}x = \frac{2}{4}x \]
Now we can substitute that back into the equation:
\[ _ = \frac{2}{4}x - \frac{5}{4}x = \frac{2 - 5}{4}x = \frac{-3}{4}x \]
Thus, the additional operation that should be included is \( -\frac{3}{4}x \).
The correct answer is:
C. \(-\frac{3}{4}x\)
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