If $x$ and $y$ are negative, then which of the following statements is/are always true? I. $x + y$ is positive. II. $xy$ is positive. III. $x - y$ is positive.

### Concept & Rule The problem evaluates the fundamental rules of operations on negative numbers. The rules state: Adding two negatives yields a negative, and multiplying two negatives yields a positive. ### Step-by-Step Solution * **Given:** $x < 0$ and $y < 0$. * Let's evaluate each statement logically: * **Statement I ($x + y$ is positive):** The sum of two negative numbers is always a larger negative number. For example, $(-2) + (-3) = -5$. This statement is strictly false. * **Statement II ($xy$ is positive):** The product of two negative numbers is always positive. $(- \times - = +)$. For example, $(-2) \times (-3) = 6$. This statement is always true. * **Statement III ($x - y$ is positive):** This expression translates to $x + (-y)$. Since $y$ is negative, $-y$ becomes positive. We are adding a negative ($x$) and a positive ($-y$). The sign of the result depends entirely on which absolute value is larger. If $x = -5, y = -2 \implies -5 - (-2) = -3$ (Negative). If $x = -2, y = -5 \implies -2 - (-5) = 3$ (Positive). Thus, it is not *always* true. ### Exam Strategy & Shortcut Memorize the multiplication parity rule: $Negative \times Negative = Positive$. Statement II is a direct definitional law of mathematics. Recognizing this instantly confirms II is true, and basic mental math verifies the others fail under certain conditions. ### Common Pitfall A frequent mistake on Statement III is assuming $x - y$ must be positive because subtracting a negative creates a positive value. Students forget that $x$ itself is negative and might "outweigh" the added positive amount. ### Final Answer Therefore, the correct answer is **II only**.