To analyze the positions of \(n\) and \(m\) on the number line described, we first need to identify their locations based on the specific definitions given.
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The point \(n\) is halfway between 0 and the first segment to the left of 0. The first segment to the left of 0 is -1. Since \(n\) is halfway between 0 and -1, we can calculate: \[ n = \frac{0 + (-1)}{2} = -\frac{1}{2} \]
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The point \(m\) is halfway between the third and fourth segments to the right of 0. The third segment is 2, and the fourth segment is 3. Since \(m\) is halfway between 2 and 3, we can calculate: \[ m = \frac{2 + 3}{2} = \frac{5}{2} \]
Now we can evaluate each of the statements:
A. The value of \(n - m\) is positive. \[ n - m = -\frac{1}{2} - \frac{5}{2} = -\frac{1 + 5}{2} = -\frac{6}{2} = -3 \] This is negative, so this statement is false.
B. The value of \(n + m\) is positive. \[ n + m = -\frac{1}{2} + \frac{5}{2} = \frac{-1 + 5}{2} = \frac{4}{2} = 2 \] This is positive, so this statement is true.
C. The value of \(- (n - m)\) is negative. \[
- (n - m) = -(-3) = 3 \] This is positive, so this statement is false.
D. The value of \(n + m\) is negative. We already calculated \(n + m = 2\), which is positive, so this statement is false.
Based on the evaluations, the only true statement is:
B. The value of \(n + m\) is positive.