To solve the equation \(\left|\frac{1}{2}x + \frac{2}{3}\right| - \frac{3}{4} = \frac{4}{5}\), follow these steps:

1. **Isolate the absolute value expression:**
\[
\left|\frac{1}{2}x + \frac{2}{3}\right| = \frac{4}{5} + \frac{3}{4}
\]

2. **Find a common denominator to add the fractions on the right-hand side:**
\[
\frac{4}{5} + \frac{3}{4} = \frac{16}{20} + \frac{15}{20} = \frac{31}{20}
\]

So the equation becomes:
\[
\left|\frac{1}{2}x + \frac{2}{3}\right| = \frac{31}{20}
\]

3. **Set up the two cases for the absolute value:**

Case 1: \(\frac{1}{2}x + \frac{2}{3} = \frac{31}{20}\)

Case 2: \(\frac{1}{2}x + \frac{2}{3} = -\frac{31}{20}\)

4. **Solve Case 1:**
\[
\frac{1}{2}x + \frac{2}{3} = \frac{31}{20}
\]

Subtract \(\frac{2}{3}\) from both sides:
\[
\frac{1}{2}x = \frac{31}{20} - \frac{2}{3}
\]

Find a common denominator to subtract the fractions:
\[
\frac{31}{20} - \frac{2}{3} = \frac{31}{20} - \frac{40}{60} = \frac{93}{60} - \frac{40}{60} = \frac{53}{60}
\]

So:
\[
\frac{1}{2}x = \frac{53}{60}
\]

Multiply both sides by 2:
\[
x = \frac{53}{30}
\]

5. **Solve Case 2:**
\[
\frac{1}{2}x + \frac{2}{3} = -\frac{31}{20}
\]

Subtract \(\frac{2}{3}\) from both sides:
\[
\frac{1}{2}x = -\frac{31}{20} - \frac{2}{3}
\]

Find a common denominator to subtract the fractions:
\[
-\frac{31}{20} - \frac{2}{3} = -\frac{31}{20} - \frac{40}{60} = -\frac{93}{60} - \frac{40}{60} = -\frac{133}{60}
\]

So:
\[
\frac{1}{2}x = -\frac{133}{60}
\]

Multiply both sides by 2:
\[
x = -\frac{133}{30}
\]

6. **Final solutions:**
\[
x = \frac{53}{30} \quad \text{or} \quad x = -\frac{133}{30}
\]

So, the solutions to the equation \(\left|\frac{1}{2}x + \frac{2}{3}\right| - \frac{3}{4} = \frac{4}{5}\) are:
\[
x = \frac{53}{30} \quad \text{or} \quad x = -\frac{133}{30}
\]