A custom operation @ is defined by the examples: -2@-4 = -8, -1@-8 = -8, and -1@6 = 6. Using the same rule (consistent across all examples), what is the value of -1@7 = ?

Introduction / Context:
This problem tests your ability to infer a custom operator definition from examples. The symbol @ does not have a standard meaning, so you must identify a rule that fits all provided input-output pairs and then apply it to a new case.

Given Data / Assumptions:

• -2@-4 = -8
• -1@-8 = -8
• -1@6 = 6
• Find -1@7 using the same rule.
• Assume one consistent arithmetic rule applies to all cases.

Concept / Approach:
Test simple operations: addition, subtraction, multiplication, division, and sign modifications. A good rule must satisfy every example, including mixed signs (negative with positive).

Step-by-Step Solution:
Try multiplication first: -2 * -4 = 8, but output is -8. This suggests the result might be the negative of the product. Define: a@b = -(a * b). Verify with -2@-4: a * b = (-2) * (-4) = 8 -(a*b) = -8, matches. Verify with -1@-8: (-1) * (-8) = 8, negative gives -8, matches. Verify with -1@6: (-1) * 6 = -6, negative gives 6, matches. Now compute -1@7: (-1) * 7 = -7 Negative of that is 7.

Verification / Alternative check:
The rule a@b = -(a*b) is validated by all three examples. Since it fits both negative-negative and negative-positive cases, applying it to -1 and 7 is consistent, giving 7.

Why Other Options Are Wrong:
-7: would be the raw product, not the negative of the product. 8 or -8: comes from misusing the earlier examples or assuming the output copies the second number. 14: would be from adding magnitudes incorrectly.

Common Pitfalls:
Assuming @ means normal multiplication, ignoring the sign flip, or fitting a rule to only one example instead of all examples are common mistakes.

Final Answer:
7