Let's solve the inequality step by step. The given inequality is:

\[
2x + \frac{3x + 1}{2} > 3x - \frac{4}{5}.
\]

1. **Eliminate the fraction:** To eliminate the fraction, we can multiply every term by \(2\) (the least common denominator):

\[
2(2x) + 2\left(\frac{3x + 1}{2}\right) > 2\left(3x\right) - 2\left(\frac{4}{5}\right).
\]

This simplifies to:

\[
4x + (3x + 1) > 6x - \frac{8}{5}.
\]

2. **Combine like terms:** On the left side, combine \(4x\) and \(3x\):

\[
4x + 3x + 1 > 6x - \frac{8}{5}.
\]

This gives:

\[
7x + 1 > 6x - \frac{8}{5}.
\]

3. **Isolate \(x\):** Next, subtract \(6x\) from both sides:

\[
7x - 6x + 1 > -\frac{8}{5}
\]

which simplifies to:

\[
x + 1 > -\frac{8}{5}.
\]

4. **Subtract 1 from both sides:** To isolate \(x\), subtract \(1\) from both sides:

\[
x > -\frac{8}{5} - 1.
\]

5. **Convert \(1\) to a fraction:** Convert \(1\) to a fraction with a denominator of \(5\):

\[
1 = \frac{5}{5}.
\]

Thus, we have:

\[
x > -\frac{8}{5} - \frac{5}{5} = -\frac{13}{5}.
\]

So, the solution to the inequality is:

\[
x > -\frac{13}{5}.
\]

This means \(x\) can take any value greater than \(-\frac{13}{5}\).