To determine which operator can be inserted between the two expressions \( 6 - 14 \cdot 28 - 214 \) and \( 34 + 4 \cdot 2 - 11 \), we first need to evaluate both expressions.
Let's calculate the left-hand side:
- \( 14 \cdot 28 = 392 \)
- \( 6 - 392 - 214 = 6 - 392 - 214 \)
- \( = 6 - 606 = -600 \)
So, the left-hand side evaluates to \( -600 \).
Next, let's calculate the right-hand side:
- \( 4 \cdot 2 = 8 \)
- \( 34 + 8 - 11 = 34 + 8 - 11 \)
- \( = 34 + 8 - 11 = 31 \)
So, the right-hand side evaluates to \( 31 \).
Now we have:
\[ -600 , _______ , 31 \]
To determine the relationship, we observe that \( -600 \) is indeed less than \( 31 \).
Thus, the operator that can be inserted to make the statement true is:
\[ < \]
The final statement is:
\[ 6 - 14 \cdot 28 - 214 < 34 + 4 \cdot 2 - 11 \]
So the answer is "less than" (<).