Let's first simplify the expressions on both sides of the inequality.
Starting with the left side: \[ 6 - 1/4 \times 28 - 2 \frac{1}{4} \]
First, convert \( 2 \frac{1}{4} \) to an improper fraction: \[ 2 \frac{1}{4} = \frac{9}{4} \]
Now, calculate \( 1/4 \times 28 \): \[ 1/4 \times 28 = 7 \]
Now, substitute this back into the expression: \[ 6 - 7 - \frac{9}{4} \]
To combine \( 6 - 7 \): \[ 6 - 7 = -1 \]
So, we have: \[ -1 - \frac{9}{4} \] Convert \(-1\) to a fraction: \[ -1 = -\frac{4}{4} \] Now combine: \[ -\frac{4}{4} - \frac{9}{4} = -\frac{4 + 9}{4} = -\frac{13}{4} \]
Now we are ready to simplify the right side: \[ \frac{3}{4} + 4 \times 2 - 11 \] Calculate \( 4 \times 2 \): \[ 4 \times 2 = 8 \]
Now rewrite the right side: \[ \frac{3}{4} + 8 - 11 \] Combine \( 8 - 11 \): \[ 8 - 11 = -3 \] So, that gives us: \[ \frac{3}{4} - 3 \] Convert \(-3\) to a fraction: \[ -3 = -\frac{12}{4} \] Now combine the fractions: \[ \frac{3}{4} - \frac{12}{4} = \frac{3 - 12}{4} = \frac{-9}{4} \]
Now we have: Left side = \(-\frac{13}{4}\) and Right side = \(-\frac{9}{4}\)
Now we can compare the two results: \[ -\frac{13}{4} \quad \text{and} \quad -\frac{9}{4} \]
This means: \[ -\frac{13}{4} < -\frac{9}{4} \]
Therefore, the appropriate operator to insert is: \[ < \] Thus, the final statement is: \[ 6 - 1/4 \times 28 - 2 \frac{1}{4} < \frac{3}{4} + 4 \times 2 - 11 \]
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