On the number line, the distance between the point whose coordinate is s and the point whose coordinate is t is greater than 500. Which of the following must be true?
I. `|s|.|t| gt 500`
II. `|s-t| gt 500`
III. `t-s gt 500`

Options

A. I only
B. II only
C. I and II only
D. I, II, and III

Answer

To solve the problem, we need to analyze the given statements based on the condition that the distance between the points \( s \) and \( t \) on the number line is greater than 500.

### Step-by-Step Solution:

1. **Understanding the Distance Condition**:
The distance between two points \( s \) and \( t \) on a number line is given by the absolute value of their difference:
\[ |s - t| > 500 \]
This means that the distance between \( s \) and \( t \) must be greater than 500.

2. **Analyzing the Statements**:
Now, let's evaluate each of the statements provided:

**Statement I**: \( |s| \cdot |t| > 500 \)
- This statement is not necessarily true. For example, if \( s = 1 \) and \( t = 600 \), then \( |s| \cdot |t| = 1 \cdot 600 = 600 > 500 \) holds true. However, if \( s = 0 \) and \( t = 600 \), then \( |s| \cdot |t| = 0 \cdot 600 = 0 \), which does not satisfy the condition. Thus, Statement I is **not necessarily true**.

**Statement II**: \( |s - t| > 500 \)
- This statement is directly derived from the distance condition we established. Since it is given that the distance between \( s \) and \( t \) is greater than 500, this statement is **true**.

**Statement III**: \( t - s > 500 \)
- This statement assumes a specific order of \( s \) and \( t \). If \( s < t \), then \( t - s > 500 \) would be true. However, if \( s > t \), then \( s - t > 500 \) would hold instead. Therefore, Statement III is **not necessarily true** as it does not account for both scenarios.

3. **Conclusion**:
Based on the evaluations:
- Statement I is not necessarily true.
- Statement II is true.
- Statement III is not necessarily true.

Thus, the only statement that must be true is Statement II.

### Final Answer:
The answer is **II only**.

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