Let's solve the equation step by step to simplify and understand it:

Given:
\[ \frac{2}{5}x - \left(-\frac{3}{5}x + \frac{1}{2}\right) = \left(\frac{3}{2}\right)^{\frac{22}{5}} \]

It seems there is an error in your equation because it appears you repeated the left-hand side with a minor change or intended for another form in the equation. To make progress, I'll assume you wanted to work with:
\[ \frac{2}{5}x - \left(-\frac{3}{5}x + \frac{1}{2}\right) = \left(\frac{3}{2}\right)^2 \]

Let's simplify the left-hand side first:
\[ \frac{2}{5}x - (-\frac{3}{5}x + \frac{1}{2}) = \frac{2}{5}x + \frac{3}{5}x - \frac{1}{2} \]

Combine the terms involving \(x\):
\[ \left(\frac{2}{5} + \frac{3}{5}\right)x - \frac{1}{2} = \frac{5}{5}x - \frac{1}{2} = x - \frac{1}{2} \]

So, the simplified left-hand side is:
\[ x - \frac{1}{2} \]

Now, working with the right-hand side with the assumption:
\[ \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

Set the simplified left-hand side equal to the right-hand side:
\[ x - \frac{1}{2} = \frac{9}{4} \]

Solve for \(x\):
1. Add \(\frac{1}{2}\) to both sides to isolate \(x\):
\[ x = \frac{9}{4} + \frac{1}{2} \]

2. Convert \(\frac{1}{2}\) to a common denominator with \(\frac{9}{4}\):
\[ \frac{1}{2} = \frac{2}{4} \]

3. Add the fractions:
\[ x = \frac{9}{4} + \frac{2}{4} = \frac{11}{4} \]

Thus, the solution for \(x\) is:
\[ x = \frac{11}{4} \]