The minimum value of the expression `abs(17x-8)-9` is

Options

A. 0
B. `-9`
C. `8/17`
D. None of these

Answer

To find the minimum value of the expression \( \text{abs}(17x - 8) - 9 \), we can follow these steps:

### Step 1: Define the function
Let \( f(x) = \text{abs}(17x - 8) - 9 \).

### Step 2: Analyze the absolute value
The expression \( \text{abs}(17x - 8) \) is always non-negative. Therefore, the minimum value of \( \text{abs}(17x - 8) \) is 0. This occurs when \( 17x - 8 = 0 \).

### Step 3: Solve for \( x \)
Set the inside of the absolute value to zero:
\[ 17x - 8 = 0 \]
Solving for \( x \):
\[ 17x = 8 \implies x = \frac{8}{17} \]

### Step 4: Calculate \( f(x) \) at \( x = \frac{8}{17} \)
Now, substitute \( x = \frac{8}{17} \) back into the function:
\[ f\left(\frac{8}{17}\right) = \text{abs}(17 \cdot \frac{8}{17} - 8) - 9 \]
This simplifies to:
\[ f\left(\frac{8}{17}\right) = \text{abs}(8 - 8) - 9 = \text{abs}(0) - 9 = 0 - 9 = -9 \]

### Step 5: Conclusion
Thus, the minimum value of the expression \( \text{abs}(17x - 8) - 9 \) is:
\[ \boxed{-9} \]