The problem is to simplify the expression $\frac{9(1+2x)}{2} - 3(3x - \frac{1}{2})$.

1. Problem Description

The problem is to simplify the expression $\frac{9(1+2x)}{2} - 3(3x - \frac{1}{2})$.

2. Solution Steps

First, distribute the 9 in the first term:

$\frac{9(1+2x)}{2} = \frac{9+18x}{2}$.

Next, distribute the -3 in the second term:

$-3(3x - \frac{1}{2}) = -9x + \frac{3}{2}$.

So the expression becomes:

$\frac{9+18x}{2} - 9x + \frac{3}{2}$.

We can rewrite the expression so all terms have a common denominator of 2:

$\frac{9+18x}{2} - \frac{18x}{2} + \frac{3}{2}$.

Combine the terms:

$\frac{9+18x - 18x + 3}{2} = \frac{12}{2}$.

Simplify the fraction:

$\frac{12}{2} = 6$.

3. Final Answer

The final answer is 6.