Let a and b be numbers such that `-a lt b +1 lt a.` Which of the following must be true ?
I. `a gt 0`
II. `|b| lt a`
III. `b gt a +1`

Options

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

Explanation

To solve the inequality given in the question, we start with the expression:

\[ -a < b + 1 < a \]

This compound inequality can be broken down into two separate inequalities:

1. \(-a < b + 1\)
2. \(b + 1 < a\)

### Step 1: Solve the first inequality

From the first inequality \(-a < b + 1\), we can isolate \(b\):

\[ b > -a - 1 \]

### Step 2: Solve the second inequality

From the second inequality \(b + 1 < a\), we can also isolate \(b\):

\[ b < a - 1 \]

### Step 3: Combine the results

Now we have:

\[ -a - 1 < b < a - 1 \]

### Step 4: Analyze the bounds for \(b\)

From the inequalities, we can see that \(b\) is bounded between \(-a - 1\) and \(a - 1\).

### Step 5: Determine the implications for \(a\)

Since \(b\) is between these two bounds, we can analyze the implications for \(a\):

1. For \(b\) to have valid values, the lower bound must be less than the upper bound:
\[ -a - 1 < a - 1 \]
Simplifying this gives:
\[ -a < a \implies 0 < 2a \implies a > 0 \]
Thus, we conclude that \(a > 0\).

### Step 6: Analyze the absolute value of \(b\)

Next, we need to check the second statement: \(|b| < a\).

Since \(b\) is bounded by \(-a - 1\) and \(a - 1\), we can analyze the maximum possible values for \(|b|\):

- The maximum value of \(b\) is \(a - 1\), and since \(a > 0\), we can conclude:
\[ |b| < a \text{ is true if } b < a \text{ and } b > -a. \]

Thus, \(|b| < a\) holds true.

### Step 7: Analyze the third statement

Now we check the third statement: \(b > a + 1\).

From the bounds we derived earlier:
\[ b < a - 1 \]
This implies that \(b\) cannot be greater than \(a + 1\) because \(a - 1 < a + 1\). Therefore, this statement is false.

### Conclusion

Based on our analysis, we conclude:

- I. \(a > 0\) is **true**.
- II. \(|b| < a\) is **true**.
- III. \(b > a + 1\) is **false**.

Thus, the correct options are I and II.