If `4 le x le 9`, then

Options

A. `(x-4)(x-9) le 0`
B. `(x-4)(x-9) ge 0`
C. `(x-4)(x-9) lt 0`
D. `(x-4)(x-9) gt 0`

Answer

To solve the problem, we need to analyze the function \( f(x) = (x - 4)(x - 9) \) and determine the sign of this function based on the given range \( 4 \leq x \leq 9 \).

### Step-by-Step Solution:

1. **Identify the function**:
We define the function as:
\[ f(x) = (x - 4)(x - 9) \]

2. **Find the critical points**:
The critical points occur when \( f(x) = 0 \):
\[ (x - 4)(x - 9) = 0 \]
This gives us:
\[ x = 4 \quad \text{and} \quad x = 9 \]

3. **Determine the intervals**:
We will analyze the sign of \( f(x) \) in the intervals defined by the critical points:
- \( x < 4 \)
- \( 4 \leq x \leq 9 \)
- \( x > 9 \)

4. **Test the interval \( x < 4 \)**:
Choose a test point, for example, \( x = 3 \):
\[ f(3) = (3 - 4)(3 - 9) = (-1)(-6) = 6 \quad (\text{positive}) \]
Thus, for \( x < 4 \), \( f(x) > 0 \).

5. **Test the interval \( 4 \leq x \leq 9 \)**:
Choose a test point, for example, \( x = 6 \):
\[ f(6) = (6 - 4)(6 - 9) = (2)(-3) = -6 \quad (\text{negative}) \]
At the endpoints:
\[ f(4) = (4 - 4)(4 - 9) = 0 \quad (\text{zero}) \]
\[ f(9) = (9 - 4)(9 - 9) = 0 \quad (\text{zero}) \]
Thus, for \( 4 < x < 9 \), \( f(x) < 0 \) and at \( x = 4 \) and \( x = 9 \), \( f(x) = 0 \).

6. **Test the interval \( x > 9 \)**:
Choose a test point, for example, \( x = 10 \):
\[ f(10) = (10 - 4)(10 - 9) = (6)(1) = 6 \quad (\text{positive}) \]
Thus, for \( x > 9 \), \( f(x) > 0 \).

7. **Conclusion**:
From our analysis, we find:
- \( f(x) > 0 \) for \( x < 4 \)
- \( f(x) = 0 \) for \( x = 4 \) and \( x = 9 \)
- \( f(x) < 0 \) for \( 4 < x < 9 \)
- \( f(x) > 0 \) for \( x > 9 \)

Therefore, the inequality that holds true for \( 4 \leq x \leq 9 \) is:
\[ f(x) \leq 0 \]
Hence, the correct option is:
\[ f(x) = (x - 4)(x - 9) \leq 0 \]