Let's evaluate both sides of the expression step by step to find a suitable operation for the blank.
First, simplify the left side:
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Start with \( 6 - \frac{1}{4} \cdot 28 - 2 \frac{1}{4} \).
- Calculate \( \frac{1}{4} \cdot 28 = 7 \).
- Convert \( 2 \frac{1}{4} \) to an improper fraction: \( 2 \frac{1}{4} = \frac{9}{4} \) (because \( 2 \cdot 4 + 1 = 9 \)).
Now we have: \[ 6 - 7 - \frac{9}{4} \]
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Convert \( 6 \) to a fraction with a denominator of 4: \[ 6 = \frac{24}{4} \]
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Now the expression becomes: \[ \frac{24}{4} - 7 - \frac{9}{4} = \frac{24}{4} - \frac{28}{4} - \frac{9}{4} \]
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Combine the fractions: \[ \frac{24 - 28 - 9}{4} = \frac{-13}{4} \]
Now let's simplify the right side:
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\( \frac{3}{4} + 4 \cdot 2 - 11 \).
- First calculate \( 4 \cdot 2 = 8 \).
- Therefore, it becomes: \[ \frac{3}{4} + 8 - 11 \]
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Combining \( 8 - 11 \): \[ 8 - 11 = -3 \]
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Now convert \( -3 \) to a fraction with a denominator of 4: \[ -3 = \frac{-12}{4} \]
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Now the right side can be evaluated: \[ \frac{3}{4} - \frac{12}{4} \] which becomes: \[ \frac{3 - 12}{4} = \frac{-9}{4} \]
Now we have:
- Left side: \( \frac{-13}{4} \)
- Right side: \( \frac{-9}{4} \)
So, we want to make \( \frac{-13}{4} \) equal to \( \frac{-9}{4} \) by placing some operation in the blank.
Now we can identify the operation.
To equate \(\frac{-13}{4}\) and \(\frac{-9}{4}\), we need:
\[ \frac{-13}{4} ; __ ; \frac{-9}{4} = 0 \]
The operation to achieve that is:
\[ \frac{-13}{4} + \frac{4}{4} = \frac{-9}{4} \]
Thus, we can place a plus (+) in the blank:
So the complete expression would be:
\( 6 - \frac{1}{4} \cdot 28 - 2 \frac{1}{4} \) + \( \frac{3}{4} + 4 \cdot 2 - 11 \)
Conclusively, the operation that fits the blank to make the statement true is: + (plus).