Which of the following describes the sum of p + (-q) where p and q are positive numbers?

Question

Which of the following describes the sum of p + (-q) where p and q are positive numbers?
The sum p + (-q) is the number located a distance |p| from q in the positive direction
The sum p + (-q) is the number located a distance |q| from p in the negative direction
The sum p + (-q) is the number located a distance |p| from q in the negative direction
The sum p + (-q) is the number located a distance |q| from p in the positive direction

GPT-4o mini · Answer

To analyze the expression $ p + (-q) $, we can rewrite it as $ p - q $. Here, both $ p $ and $ q $ are positive numbers. 1. If $ p > q $, then $ p - q $ is a positive number, meaning you can visualize this as moving $ q $ units to the left from $ p $. 2. If $ p < q $, then $ p - q $ is a negative number, meaning you can visualize this as moving $ |q| $ units to the left of $ |p| $. Now, let’s evaluate each option based on this understanding: 1. **The sum $ p + (-q) $ is the number located a distance $|p|$ from $ q $ in the positive direction**: This is incorrect since the distance is not from $ q $ to $ p $ in this way. 2. **The sum $ p + (-q) $ is the number located a distance $|q|$ from $ p $ in the negative direction**: This is also misleading; it suggests that you would be at $ p $ and then would move down $ |q| $ units, which is not a correct characterization depending on the relationship between $ p $ and $ q $. 3. **The sum $ p + (-q) $ is the number located a distance $|p|$ from $ q $ in the negative direction**: This is not correct because it describes moving from $ q $ toward a point $ p $, which could misrepresent the result. 4. **The sum $ p + (-q) $ is the number located a distance $|q|$ from $ p $ in the positive direction**: This statement makes an implication that wouldn’t normally hold under subtraction and needs to be evaluated for sign differences. Ultimately, we can conclude that the correct interpretation of $ p + (-q) $ must acknowledge the basic arithmetic properties—primarily subtraction and the direction as either positive or negative. Thus, the most appropriate characterization (if forced to select from the given options) might be the sum that best resembles the movement of subtracting $ q $ from $ p $, recognizing the principle of distance. But strictly speaking, none of the options perfectly convey what occurs with $ p - q $. Given the relationships and understanding of mathematical representations, the closest correct interpretation would have to integrate `p being less than q` or `more than q`. Unfortunately, based on the clarity you require and the choices you've given, none of those listed correctly describe the mathematical operation $ p + (-q) $ uniformly. If an additional choice were to represent $ p $ minus $ q $ correctly in terms of distance and direction, that would be preferred.