Question

Consider the following expressions: 6−14⋅28−2146−14⋅28−214 _______ 34+4⋅2−1134+4⋅2−11 . Which operator can be inserted to make the statement true?

Answer 1

To determine which operator can be inserted in the blank of the expression

\[ 6 - 14 \cdot 28 - 2146 - 14 \cdot 28 - 214 \, _______ \, 34 + 4 \cdot 2 - 11 \]

and make it a true statement, we should first evaluate each side of the equation without the blank operator.

1. **Calculate the left-hand side (LHS)**:
\[
6 - 14 \cdot 28 - 214
\]
First, calculate \( 14 \cdot 28 \):
\[
14 \cdot 28 = 392
\]
Then substitute this value into the left-hand side:
\[
6 - 392 - 214
\]
Next, combine \( 6 - 392 \):
\[
6 - 392 = -386
\]
Finally, calculate \( -386 - 214 \):
\[
-386 - 214 = -600
\]
Hence,
\[
LHS = -600
\]

2. **Calculate the right-hand side (RHS)**:
\[
34 + 4 \cdot 2 - 11
\]
First, calculate \( 4 \cdot 2 \):
\[
4 \cdot 2 = 8
\]
Then substitute this value into the right-hand side:
\[
34 + 8 - 11
\]
Next, combine \( 34 + 8 \):
\[
34 + 8 = 42
\]
Finally, calculate \( 42 - 11 \):
\[
42 - 11 = 31
\]
Hence,
\[
RHS = 31
\]

3. **Determine the operator**:
Now we have:
\[
-600 \, _______ \, 31
\]
We need to find an operator that makes this statement true. Let's evaluate common operators:

- **Addition**: \(-600 + 31 = -569\) (not true)
- **Subtraction**: \(-600 - 31 = -631\) (not true)
- **Multiplication**: \(-600 \times 31 = -18600\) (not true)
- **Division**: \(-600 \div 31 \approx -19.35\) (not true)
- **Equals**: \(-600 = 31\) (not true)

Since none of the standard arithmetic operations make the statement true, it's helpful to consider that we are working with a comparison.

In conclusion, the operator that can be inserted to make either side valid under comparison is:
\[
\text{< (less than)}
\]

Thus, the complete expression is:
\[
6 - 14 \cdot 28 - 214 < 34 + 4 \cdot 2 - 11
\]
This evaluates to:
\[
-600 < 31
\]
Which is true. Therefore, the inserted operator is:

\(\boxed{<}\)