To solve the problem, we have the inequality given as:
\[
-y < x < y
\]
We need to analyze the three statements to determine which must be true based on this inequality.
### Step 1: Analyze the inequality
From the inequality \(-y < x < y\), we can derive some information about \(x\) and \(y\).
1. The left part of the inequality \(-y < x\) implies that \(x\) is greater than \(-y\).
2. The right part of the inequality \(x < y\) implies that \(x\) is less than \(y\).
### Step 2: Check the first statement: \(|x| < y\)
To check if \(|x| < y\) must be true, we can analyze the bounds on \(x\):
- Since \(x < y\), we know that \(x\) cannot be equal to or greater than \(y\).
- If \(x\) is negative (which it can be, since \(x\) can be less than \(y\)), then \(|x| = -x\).
Now, since \(-y < x < y\), let's consider the case where \(x\) is negative.
- If \(x\) is negative, then \(|x| = -x\) and since \(-y < x\), we can say that \(-x < y\). Thus, \(|x| < y\) holds true.
If \(x\) is positive, then \(|x| = x\) and since \(x < y\), it also holds that \(|x| < y\).
Thus, the first statement is true.
### Step 3: Check the second statement: \(x > 0\)
The inequality \(-y < x < y\) does not guarantee that \(x\) is positive.
- For example, if we take \(y = 3\) and \(x = -2\), then \(-3 < -2 < 3\) holds true, but \(x\) is not greater than 0.
Thus, the second statement is not necessarily true.
### Step 4: Check the third statement: \(y > 0\)
The inequality \(-y < x < y\) implies that \(y\) must be positive.
- If \(y\) were negative or zero, then \(-y\) would be positive or zero, which would contradict the inequality \(-y < x\) because \(x\) would have to be less than or equal to zero in that case.
Thus, \(y\) must be greater than 0.
### Conclusion
Based on the analysis, we conclude that:
- Statement I: \(|x| < y\) is true.
- Statement II: \(x > 0\) is not necessarily true.
- Statement III: \(y > 0\) is true.
Therefore, the statements that must be true are I and III.
### Final Answer
The statements that must be true are:
- I. \(|x| < y\)
- III. \(y > 0\)
