To make the expression \( \frac{5}{4}x + 6 + ____ \) equivalent to \( \frac{1}{2}x + 6 \), we should first simplify the comparison.
- Rewrite \( \frac{1}{2} \) in terms of fourths: \[ \frac{1}{2} = \frac{2}{4} \]
Now, we can set up the equation: \[ \frac{5}{4}x + 6 + ___ = \frac{2}{4}x + 6 \]
Next, to make the left side equal to the right side, we need to remove \( \frac{5}{4}x \) and only have \( \frac{2}{4}x \) left.
So, we need to add something that will effectively subtract \( \frac{5}{4}x \) and leave \( \frac{2}{4}x \): \[ ___ = \frac{2}{4}x - \frac{5}{4}x \]
Calculating this gives: \[ ___ = \frac{2}{4}x - \frac{5}{4}x = -\frac{3}{4}x \]
Thus, the additional operation that should be included in the expression to make it equivalent to \( \frac{1}{2}x + 6 \) is:
−3/4x
So the correct response is −3/4x.