To evaluate the expression \( 6 - \frac{1}{4} \cdot 28 - 2 \frac{1}{4} \) and \( \frac{3}{4} + 4 \cdot 2 - 11 \), we need to simplify both sides.
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Simplifying the left side: \[ 6 - \frac{1}{4} \cdot 28 - 2 \frac{1}{4} \] First, calculate \( \frac{1}{4} \cdot 28 \): \[ \frac{1}{4} \cdot 28 = 7 \] Then, we convert \( 2 \frac{1}{4} \) to an improper fraction: \[ 2 \frac{1}{4} = \frac{9}{4} \] Now, substituting these into the expression: \[ 6 - 7 - \frac{9}{4} \] First, calculate \( 6 - 7 \): \[ 6 - 7 = -1 \] Now, converting \(-1\) to a fraction: \[ -1 = \frac{-4}{4} \] Thus, we have: \[ -1 - \frac{9}{4} = \frac{-4}{4} - \frac{9}{4} = \frac{-13}{4} \]
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Simplifying the right side: \[ \frac{3}{4} + 4 \cdot 2 - 11 \] First, calculate \( 4 \cdot 2 \): \[ 4 \cdot 2 = 8 \] Now substitute: \[ \frac{3}{4} + 8 - 11 \] Calculate \( 8 - 11 \): \[ 8 - 11 = -3 \] Again, converting \(-3\) to a fraction: \[ -3 = \frac{-12}{4} \] Thus, we have: \[ \frac{3}{4} - 3 = \frac{3}{4} + \frac{-12}{4} = \frac{3 - 12}{4} = \frac{-9}{4} \]
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Now we can compare the two results: \[ \frac{-13}{4} \quad \text{and} \quad \frac{-9}{4} \]
Since \(\frac{-13}{4} < \frac{-9}{4}\), we determine that the left side is less than the right side.
Therefore, the correct operator to use is:
\[ < \quad \text{(less than)} \]