Let A = \{x \in \mathbb{R} / [x + 3] + [x + 5] <= 3\} B = \{x \in \mathbb{R} / (3 ^ x)|x + 3| < 1\} , where [t] denotes greatest integer less than or equal to t. Then,

(1) A cap B = phi

(2) A = B

(3) B subset A

(4) A subset B

To solve the given problem, let's break down the sets \( A \) and \( B \) one by one and analyze their properties.

### Set \( A \):
\[ A = \{x \in \mathbb{R} \mid [x + 3] + [x + 5] \leq 3\} \]
Here, \( [t] \) denotes the greatest integer less than or equal to \( t \).

1. **Understanding the expression \( [x + 3] + [x + 5] \):**
   - The greatest integer function \( [x + 3] \) and \( [x + 5] \) represent the greatest integers less than or equal to \( x + 3 \) and \( x + 5 \), respectively.

2. **Range Analysis:**
   - If \( x \) is an integer, then \( [x + 3] \) and \( [x + 5] \) are simply \( x + 3 \) and \( x + 5 \).
   - Consider non-integer \( x \) in intervals \( n \leq x < n+1 \) where \( n \) is an integer.

3. **Considering specific values of \( x \):**
   - For \( x = 0 \): \( [3] + [5] = 3 + 5 = 8 \) (not possible as \( 8 > 3 \))
   - For \( x = -3.5 \): \( [x + 3] = [0.5] = 0 \), \( [x + 5] = [1.5] = 1 \), hence \( 0 + 1 = 1 \leq 3 \).

4. **Possible values for \( x \):**
   - We must have \( [x + 3] + [x + 5] \leq 3 \), so \( x \) must be around intervals where \( x+3 \) and \( x+5 \) are near the integer values of 0, 1, 2, etc.

5. **Conclude \( A \):**
   - The values of \( x \) that satisfy this inequality are \( -3.5 \leq x < -2 \) or \( -3 \leq x < -2.5 \).

### Set \( B \):
\[ B = \{x \in \mathbb{R} \mid (3^x) \cdot |x + 3| < 1 \} \]

1. **Analyzing \( (3^x) \cdot |x + 3| < 1 \):**
   - \( 3^x \) is an exponential function that grows rapidly.
   - The inequality implies \( 3^x < \frac{1}{|x + 3|} \).

2. **Behavior based on \( x \):**
   - If \( x \) is very negative, \( 3^x \) becomes small, and the inequality might be satisfied.
   - If \( x \) is close to \( -3 \), then \( |x + 3| \) is small, making \( \frac{1}{|x + 3|} \) large, possibly satisfying the inequality.

3. **Exact Solutions:**
   - For \( x = -3 \), \( |x + 3| = 0 \), which does not satisfy the inequality (division by zero).
   - Consider small deviations around \( x = -3 \).

### Intersection and Subset Analysis:
- **\( A \cap B \):** 
   - There might be a common interval where both inequalities are satisfied.
   - \( A \) includes values like \( x \) near \( -3.5 \), which can be verified for \( B \).

- **Subset Relationships:**
   - If all elements of \( B \) are within \( A \), \( B \subset A \).
   - Conversely, \( A \subset B \) might be true if all elements of \( A \) satisfy the inequality for \( B \).

### Conclusion:
After analyzing the behavior and checking possible values of \( x \), the best fit is that \( B \subset A \).

Thus, the correct answer is \( \boxed{3} \) \( B \subset A \).

Would you like more details or have any questions? Here are some related questions:

1. How does the greatest integer function behave for non-integer inputs?
2. What are the general properties of exponential functions like \( 3^x \)?
3. How does the condition \( |x+3| \) affect the inequality in set \( B \)?
4. What happens if \( x \) approaches \( -3 \) from the left or right?
5. Can the behavior of \( B \) change if \( 3^x \) were replaced with another base?

**Tip:** Always consider the specific intervals where the greatest integer function changes its value when solving inequalities.

Explore how to solve inequalities involving set theory, greatest integer functions, and exponential functions. Analyze sets A and B defined with mathematical precision. Suitable for undergraduate level students and enthusiasts of mathematical concepts involving set theory and functions.

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