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.
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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, \[ LHS = -600 \]
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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, \[ RHS = 31 \]
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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:
\(\boxed{<}\)
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