If `4 le x le 9` then the expression (x-4) (x-9) is

Options

A. `ge 0`
B. `le 0`
C. `gt 0`
D. `lt 0`

Answer

To solve the problem, we need to analyze the expression \((x - 4)(x - 9)\) given the constraint \(4 \leq x \leq 9\).

### Step-by-Step Solution:

1. **Identify the expression**: We have the expression \((x - 4)(x - 9)\).

2. **Evaluate the expression at the endpoints**:
- When \(x = 4\):
\[ (4 - 4)(4 - 9) = 0 \cdot (-5) = 0 \]
- When \(x = 9\):
\[ (9 - 4)(9 - 9) = 5 \cdot 0 = 0 \]

3. **Evaluate the expression at a point between 4 and 9**:
- Let's choose \(x = 5\):
\[ (5 - 4)(5 - 9) = 1 \cdot (-4) = -4 \]

4. **Analyze the sign of the expression**:
- For \(x = 4\) and \(x = 9\), the expression equals 0.
- For \(x = 5\), the expression is negative.
- Since the expression \((x - 4)(x - 9)\) is a quadratic function that opens upwards (the coefficient of \(x^2\) is positive), it will be negative between the roots \(4\) and \(9\).

5. **Conclusion**:
- Therefore, for \(4 < x < 9\), the expression \((x - 4)(x - 9)\) is negative.
- At the endpoints \(x = 4\) and \(x = 9\), the expression is equal to 0.

### Final Result:
The expression \((x - 4)(x - 9)\) is less than or equal to 0 for \(4 \leq x \leq 9\).