To solve the expression \(4.5 \cdot \frac{3}{2} - \frac{9}{7} \cdot \frac{1}{24.5} \cdot \frac{3}{2} - \frac{9}{7} \cdot \frac{1}{2}\), let's break it down step by step.

### Step 1: Calculate \(4.5 \cdot \frac{3}{2}\)

\[
4.5 \cdot \frac{3}{2} = \frac{4.5 \cdot 3}{2} = \frac{13.5}{2} = 6.75
\]

### Step 2: Calculate \(\frac{9}{7} \cdot \frac{1}{24.5} \cdot \frac{3}{2}\)

First, calculate \(\frac{1}{24.5}\):
\[
24.5 = \frac{245}{10} = \frac{49}{2} \quad \text{(So, } \frac{1}{24.5} = \frac{2}{49} \text{)}
\]

Now calculate:
\[
\frac{9}{7} \cdot \frac{2}{49} \cdot \frac{3}{2} = \frac{9 \cdot 2 \cdot 3}{7 \cdot 49 \cdot 2}
\]

The \(2\) in the numerator and the denominator cancels out:
\[
= \frac{9 \cdot 3}{7 \cdot 49} = \frac{27}{343}
\]

### Step 3: Calculate \(\frac{9}{7} \cdot \frac{1}{2}\)

\[
\frac{9}{7} \cdot \frac{1}{2} = \frac{9}{14}
\]

### Step 4: Combine all parts together

Now we can rewrite the original expression as:
\[
6.75 - \frac{27}{343} - \frac{9}{14}
\]

### Step 5: Convert all terms to a common denominator

The least common multiple of \(343\) and \(14\) is \(343\).

Convert \(\frac{9}{14}\) to have a denominator of \(343\):
\[
\frac{9}{14} = \frac{9 \cdot 24.5}{14 \cdot 24.5} = \frac{220.5}{343}
\]

Now, convert \(6.75\):
\[
6.75 = \frac{6.75 \cdot 343}{343}
\]
Calculating \(6.75 \cdot 343\):
\[
6.75 \cdot 343 = 2317.25 \quad \text{(So, } 6.75 = \frac{2317.25}{343} \text{)}
\]

### Step 6: Now combine everything

\[
\frac{2317.25}{343} - \frac{27}{343} - \frac{220.5}{343}
= \frac{2317.25 - 27 - 220.5}{343}
= \frac{2070.75}{343}
\]

Thus, the final result of the entire expression \(4.5 \cdot \frac{3}{2} - \frac{9}{7} \cdot \frac{1}{24.5} \cdot \frac{3}{2} - \frac{9}{7} \cdot \frac{1}{2}\) is:
\[
\frac{2070.75}{343} \approx 6.05 \text{ (approximately rounded)}
\]