Let's analyze the expression \( p + (-q) \).
Since \( q \) is a positive number, the term \( (-q) \) represents a negative number equal in magnitude to \( q \).
The expression can be rewritten as:
\[ p + (-q) = p - q \]
Now, let's understand what \( p - q \) represents:
- If \( p - q \) is positive, it means that \( p \) is greater than \( q \), and the result is a distance of \( (p - q) \) from zero in the positive direction.
- If \( p - q \) is negative, it indicates that \( p \) is less than \( q\), and the result is a distance of \( (q - p) \) from zero in the negative direction.
- If \( p = q \), the result is zero.
Now, we look at the provided options.
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The first option states that the sum \( p + (-q) \) is the number located a distance \( |q| \) from \( p \) in the positive direction. This is not correct as it’s located in the direction determined by the sign of \( p - q \).
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The second option states that the sum \( p + (-q) \) is the number located a distance \( |q| \) from \( p \) in the negative direction. This is closer to being correct. Indeed, if you consider moving \( q \) units left from \( p \), that reflects \( p - q \).
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The third option is not applicable since it discusses being located a distance from \( q \), which doesn’t relate to \( p \) directly.
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The fourth option states a distance from \( p \) in the positive direction, which is again incorrect since the addition of negative values indicates movement in the opposite direction.
Given these analyses, the only statement that correctly describes \( p + (-q) \) is:
The sum p + (-q) is the number located a distance |q| from p in the negative direction.
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