Consider the following expressions:

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:
\[
6 - 1/4 \times 28 - 2 \frac{1}{4} < \frac{3}{4} + 4 \times 2 - 11
\]