If $p > q$ and $r < 0$, then which is true?

### Concept & Rule A fundamental property of inequalities is that multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. ### Step-by-Step Solution 1. Start with the given primary inequality: $p > q$. 2. We are given a condition: $r < 0$ (meaning $r$ is a negative number). 3. Multiply both sides of $p > q$ by $r$. 4. Because we are multiplying by a negative number, the inequality sign $>$ flips to $<$. 5. The resulting inequality is $pr < qr$. 6. Let's check the other options for validity: Adding or subtracting a number (whether positive or negative) does not flip the sign. Thus, $p - r > q - r$ and $p + r > q + r$, making options (b) and (c) false. ### Exam Strategy & Shortcut Whenever variables lack specific values in inequality problems, plug in simple numbers that fit the constraints. Let $p = 2$, $q = 1$ (so $2 > 1$). Let $r = -1$ (so $-1 < 0$). Option (a): $2(-1) < 1(-1) \Rightarrow -2 < -1$ (True). ### Common Pitfall Mechanically applying operations without considering the sign of the variable. Students often select an answer that assumes $pr > qr$, forgetting that $r$ is negative. ### Final Answer **Therefore, the correct answer is $pr < qr$.**