If `-4ltalt4 and -2 lt b lt-1`, which of the following could NOT be the value of ab?

Options

A. `-3`
B. 0
C. 4
D. 9

Answer

To solve the problem, we need to analyze the ranges of \( a \) and \( b \) and determine the possible values of their product \( ab \).

### Step-by-Step Solution:

1. **Identify the ranges of \( a \) and \( b \)**:
- From the problem, we have:
\[ -4 < a < 4 \]
\[ -2 < b < -1 \]

2. **Determine the possible range of \( ab \)**:
- Since \( a \) can take values between \(-4\) and \(4\) and \( b \) can take values between \(-2\) and \(-1\), we need to find the minimum and maximum values of the product \( ab \).
- The product \( ab \) will be negative since \( a \) can be positive or negative and \( b \) is always negative.

3. **Calculate the extreme products**:
- The maximum product occurs when \( a \) is at its maximum and \( b \) is at its minimum:
\[ \text{Max } ab = 4 \times (-1) = -4 \]
- The minimum product occurs when \( a \) is at its minimum and \( b \) is at its maximum:
\[ \text{Min } ab = -4 \times (-2) = 8 \]

4. **Conclusion on the range of \( ab \)**:
- Therefore, the possible range of \( ab \) is:
\[ -4 < ab < 8 \]

5. **Evaluate the options**:
- Now we check each option to see which one cannot be achieved within the range of \( ab \):
- Option 1: \(-3\) (possible, since \(-4 < -3 < 8\))
- Option 2: \(0\) (possible, since \(-4 < 0 < 8\))
- Option 3: \(4\) (possible, since \(-4 < 4 < 8\))
- Option 4: \(9\) (not possible, since \(9\) is greater than \(8\))

6. **Final Answer**:
- The value that could NOT be the value of \( ab \) is:
\[ \boxed{9} \]