If `9lexle16`, then

Options

A. `(x-9)(x-16)le0`
B. `(x-9)(x-16)ge0`
C. `(x-9)(x-16)lt0`
D. `(x-9)(x-16)gt0`

Answer

To solve the inequality \(9 \leq x \leq 16\) and find the value of \( (x - 9)(x - 16) \), we can follow these steps:

### Step 1: Understand the inequality
The inequality \(9 \leq x \leq 16\) means that \(x\) can take any value between 9 and 16, inclusive.

### Step 2: Identify the critical points
The critical points from the inequality are \(x = 9\) and \(x = 16\). These points will help us determine the intervals for the expression \( (x - 9)(x - 16) \).

### Step 3: Analyze the expression
The expression \( (x - 9)(x - 16) \) will change its sign at the critical points \(x = 9\) and \(x = 16\).

### Step 4: Test intervals
We need to check the sign of the expression in the intervals defined by the critical points:
1. For \(x < 9\): Choose \(x = 8\)
\[ (8 - 9)(8 - 16) = (-1)(-8) = 8 \quad (\text{positive}) \]
2. For \(9 < x < 16\): Choose \(x = 10\)
\[ (10 - 9)(10 - 16) = (1)(-6) = -6 \quad (\text{negative}) \]
3. For \(x > 16\): Choose \(x = 17\)
\[ (17 - 9)(17 - 16) = (8)(1) = 8 \quad (\text{positive}) \]

### Step 5: Determine the sign of the expression
- The expression \( (x - 9)(x - 16) \) is positive when \(x < 9\) or \(x > 16\).
- The expression is negative when \(9 < x < 16\).
- At the critical points \(x = 9\) and \(x = 16\), the expression equals zero.

### Step 6: Conclusion
Since we are interested in the values of \(x\) that satisfy the inequality \(9 \leq x \leq 16\), we find that:
- The expression \( (x - 9)(x - 16) \) is less than or equal to zero in the interval \(9 \leq x \leq 16\).

Thus, the final answer is:
\[ (x - 9)(x - 16) \leq 0 \quad \text{for } 9 \leq x \leq 16 \]