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Difficulty: MediumPositive and Negative Number Properties

Let mm and nn be integers such that −4≤m≤6-4 \le m \le 6 and −8≤n≤3-8 \le n \le 3. If m2n<0m^2 n < 0 and m+n>0m + n > 0, what is the minimum possible value of the product m⋅nm \cdot n?

Answer: -30

Answer

The minimum possible value of m⋅nm \cdot n is −30-30.
The correct answer is -30. Analyzing m2n<0m^2 n < 0 shows that nn must be negative and mm cannot be zero. The condition m+n>0m + n > 0 implies m>−n=∣n∣m > -n = |n|, making mm positive. Since m≤6m \le 6, the maximum possible value for ∣n∣|n| is 5, which corresponds to n=−5n = -5. When n=−5n = -5, mm must be strictly greater than 5, leaving m=6m = 6 as the only valid value. The product is 6×(−5)=−306 \times (-5) = -30, which is the minimum value achievable under all constraints.

Step-by-Step Solution

1
Determine the signs of mm and nn using the given inequalities.
n<0n < 0 and m>0m > 0 with m≠0m \ne 0.
Since m2m^2 is strictly positive for any non-zero integer mm, m2n<0m^2 n < 0 forces n<0n < 0. Then m+n>0m + n > 0 requires m>−n>0m > -n > 0.
2
Find the range of valid integer values for nn.
n∈{−5,−4,−3,−2,−1}n \in \{-5, -4, -3, -2, -1\}.
Since m≤6m \le 6 and m>−nm > -n, we must have −n≤5-n \le 5, which gives n≥−5n \ge -5.
3
Evaluate the minimum product m⋅nm \cdot n across all allowed values of nn.
The minimum product is −30-30, occurring when n=−5n = -5 and m=6m = 6.
To minimize a negative product, maximize the absolute product m⋅∣n∣m \cdot |n|. When n=−5n = -5, mm must be 66, yielding 6⋅(−5)=−306 \cdot (-5) = -30.

Key Concept

Positive and Negative Number Properties with Inequalities
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Topics

Positive and Negative Number Properties

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