A student claims that if x and y are integers, one positive and one negative, then the product xy must be negative. Which of the following is a correct evaluation of the student's claim?
The claim is true.
When x and y are integers, with one being positive and the other negative, their product is always negative. This is due to the fundamental properties of multiplication involving integers, where a positive number multiplied by a negative number results in a negative product.
This option accurately reflects the mathematical principle that the product of a positive integer and a negative integer yields a negative integer. Therefore, the student's assertion holds true under these conditions.
This statement is incorrect because it describes a scenario where both integers are negative. The product of two negative integers is indeed positive, but this does not pertain to the student's claim, which specifically involves one positive and one negative integer.
This choice misrepresents the situation by introducing zero as one of the integers. The product involving zero does not support the claim because zero is neither positive nor negative, thus failing to address the conditions set by the original statement.
Similar to option C, this option also involves zero, which does not meet the criteria of one positive and one negative integer. The product is zero, but this does not contradict the student's claim about the product of a positive and a negative integer.
This choice incorrectly highlights a product of a negative and a positive integer, which results in a negative product (-12). Thus, it does not demonstrate a contradiction to the student's claim; rather, it reaffirms the original assertion.
The student's claim that the product of one positive and one negative integer must be negative is indeed correct. Each of the incorrect options either misapplies the conditions set by the claim or fails to address the fundamental rule of integer multiplication. Therefore, the assertion stands validated, confirming that the product will always be negative under the specified circumstances.
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