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

Question

A. the sum p + (-q) is the number located a distance from |q| in a positive direction
B. the sum p+ (-q) is the number located a distance from |q| in a negative direction
C. the sum p+ (-q) is the number located a distance from |p| in a negative direction
D. the sum p+ (-q) is the number located a distance from |p| in a positive direction

Answer 1

To analyze the expression \( p + (-q) \), we first rewrite it as \( p - q \). Given that \( p \) and \( q \) are positive numbers, the sum \( p - q \) can be interpreted based on the relationship between \( p \) and \( q \).

1. If \( p > q \), then \( p - q \) will be positive, and will be located \( p \) units to the right of zero and \( q \) units to the left of \( p \). Hence, the result will be \( p - q \) and can be visualized on the number line.

2. If \( p < q \), then \( p - q \) will be negative, and will lie to the left of zero.

3. If \( p = q \), the result will be \( 0 \).

Now let's evaluate the options:

• A. Says "the sum \( p + (-q) \) is the number located a distance from |q| in a positive direction". This is incorrect; that would imply we are increasing \( |q| \), which is not what happens with \( p - q \).

• B. Says "the sum \( p + (-q) \) is the number located a distance from |q| in a negative direction". This could be true in the case where \( p < q \), but not necessarily true in all cases.

• C. Says "the sum \( p + (-q) \) is the number located a distance from |p| in a negative direction". This does not represent the operation of subtracting \( q \) from \( p \).

• D. Says "the sum \( p + (-q) \) is the number located a distance from |p| in a positive direction". This is incorrect as well for similar reasons to option C.

Given the analysis above, none of the options fully and accurately describes the sum \( p + (-q) \) under all conditions that can arise based on the relationship between \( p \) and \( q \).

However, focusing specifically on the absolute value and direction:

• The sum \( p - q \) is effectively the difference and can be seen as moving \( p \) and adjusting \( q \), indicating directional distance with relation to both terms.

Nonetheless, among them, option B may conceptually align under specific simple scenarios but lacks completeness across general cases.

Overall, no single choice encapsulates the expression fully without ambiguity based on the positivity of \( p \) and \( q \).