The maximum value of the expression `27-abs(9x-8)` is :

Options

A. 27
B. 17
C. 44
D. 26

Explanation

To find the maximum value of the expression \( 27 - |9x - 8| \), we can follow these steps:

### Step 1: Understand the expression
The expression we need to maximize is \( 27 - |9x - 8| \). To maximize this expression, we need to minimize the term \( |9x - 8| \).

### Step 2: Analyze the modulus function
The modulus function \( |k| \) is always non-negative, meaning \( |9x - 8| \geq 0 \) for all values of \( x \). The minimum value of \( |9x - 8| \) is 0.

### Step 3: Find when the modulus is zero
To find when \( |9x - 8| = 0 \), we set the inside of the modulus to zero:
\[ 9x - 8 = 0 \]
Solving for \( x \):
\[ 9x = 8 \implies x = \frac{8}{9} \]

### Step 4: Substitute back to find the maximum value
Now that we know the minimum value of \( |9x - 8| \) is 0 (which occurs when \( x = \frac{8}{9} \)), we can substitute this back into the original expression:
\[ 27 - |9x - 8| = 27 - 0 = 27 \]

### Conclusion
Thus, the maximum value of the expression \( 27 - |9x - 8| \) is \( 27 \).

### Final Answer
The maximum value of the expression is \( \boxed{27} \).

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