`A_(4) ={b: b in Z " and " -7 lt 3b - 1 le 2}`

Answer

To solve the problem, we need to find the set \( A_4 \) which contains integers \( b \) such that \( -7 < 3b - 1 \leq 2 \).

### Step-by-Step Solution:

1. **Set Up the Inequality**:
We start with the compound inequality:
\[ -7 < 3b - 1 \leq 2 \]

2. **Break It Down**:
We can break this into two parts:
- Part 1: \( -7 < 3b - 1 \)
- Part 2: \( 3b - 1 \leq 2 \)

3. **Solve Part 1**:
For the first part, we add 1 to both sides:
\[ -7 + 1 < 3b \implies -6 < 3b \]
Now, divide by 3:
\[ -2 < b \]
This simplifies to:
\[ b > -2 \]

4. **Solve Part 2**:
For the second part, we also add 1 to both sides:
\[ 3b - 1 \leq 2 \implies 3b \leq 2 + 1 \implies 3b \leq 3 \]
Now, divide by 3:
\[ b \leq 1 \]

5. **Combine the Results**:
From both parts, we have:
\[ -2 < b \leq 1 \]
This means \( b \) can take values greater than -2 and less than or equal to 1.

6. **Identify Integer Solutions**:
The integers that satisfy this inequality are:
- The integers greater than -2 are: -1, 0, 1
- Since \( b \) must also be less than or equal to 1, we include 1.

7. **Form the Set**:
Therefore, the set \( A_4 \) is:
\[ A_4 = \{-1, 0, 1\} \]

### Final Answer:
\[ A_4 = \{-1, 0, 1\} \]