19. 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?
Answer: A
The claim is true.
The student's assertion that if x and y are integers, one positive and one negative, then the product xy must be negative is indeed correct. When multiplying a positive integer by a negative integer, the result is always negative.
A) The claim is true.
This option accurately reflects the student's claim. When one number is positive and the other is negative, the multiplication of these two integers will always yield a negative product, confirming the validity of the claim.
B) The claim is false since (-2)(-5) = 10.
This option incorrectly argues against the student's claim by presenting the product of two negative integers. However, the student's claim specifically involves one positive and one negative integer, making this scenario irrelevant to the evaluation.
C) The claim is false since (0)(-2) = 0.
This option also misinterprets the student's claim. The product involves zero, which is neither positive nor negative, and does not address the scenario of one positive and one negative integer as specified by the student.
D) The claim is false since (0)(3) = 0.
Similar to option C, this choice involves zero and does not pertain to the student's statement regarding the multiplication of one positive integer and one negative integer. Therefore, it does not provide a valid counterexample to the claim.
E) The claim is false since (-3)(4) = 12.
This option incorrectly argues against the student's claim by showing the product of a negative integer and a positive integer. However, it mistakenly suggests a positive product, which contradicts the student's assertion and does not apply to the context of one positive and one negative integer.
Conclusion
The student's claim is definitively correct, as the product of one positive and one negative integer will always result in a negative value. All other options fail to provide a relevant counterexample since they either involve two integers of the same sign or zero, which do not pertain to the conditions set forth in the student's statement.