To make the expression \( \frac{5}{6}x + 6 + ___ \) equivalent to \( \frac{1}{2}x + 6 \), we need to determine what addition we need to add in place of the blank.
First, let's express \( \frac{1}{2} \) as a fraction with a denominator of 6 for easier comparison:
\[ \frac{1}{2} = \frac{3}{6} \]
Now we want \( \frac{5}{6}x + 6 + ___ \) to equal \( \frac{3}{6}x + 6 \).
Now, let's set up the equation:
\[ \frac{5}{6}x + 6 + __ = \frac{3}{6}x + 6 \]
To isolate the unknown, we can subtract \( 6 \) from both sides:
\[ \frac{5}{6}x + __ = \frac{3}{6}x \]
Now, subtract \( \frac{5}{6}x \) from both sides:
\[ __ = \frac{3}{6}x - \frac{5}{6}x \]
Now, perform the subtraction:
\[ __ = \frac{3 - 5}{6}x = -\frac{2}{6}x = -\frac{1}{3}x \]
So, the additional operation that should be included in the expression is:
\[ -\frac{1}{3}x \]
Therefore, the complete expression is:
\[ \frac{5}{6}x + 6 - \frac{1}{3}x \]
This will make it equivalent to \( \frac{1}{2}x + 6 \).
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