Let's solve each of the inequalities step by step:

### a) \(-\frac{4}{3}x + 4 < \frac{2}{3}\)

1. Subtract 4 from both sides:
\[
-\frac{4}{3}x < \frac{2}{3} - 4
\]

Convert 4 to a fraction:
\[
-\frac{4}{3}x < \frac{2}{3} - \frac{12}{3} \implies -\frac{4}{3}x < -\frac{10}{3}
\]

2. Multiply both sides by \(-\frac{3}{4}\) (remember to reverse the inequality):
\[
x > \frac{10}{4} \implies x > \frac{5}{2}
\]

### b) \(\frac{1}{2}x - 5 > -\frac{3}{4}x\)

1. Add \(\frac{3}{4}x\) to both sides:
\[
\frac{1}{2}x + \frac{3}{4}x > 5
\]

2. Find a common denominator (which is 4):
\[
\frac{2}{4}x + \frac{3}{4}x > 5 \implies \frac{5}{4}x > 5
\]

3. Multiply both sides by \(\frac{4}{5}\):
\[
x > 4
\]

### c) \(3 - 0.2x < 13\)

1. Subtract 3 from both sides:
\[
-0.2x < 10
\]

2. Divide by -0.2 (reverse inequality):
\[
x > -50
\]

### d) \(\frac{1}{2} + \frac{x}{3} \geq \frac{1}{4}\)

1. Subtract \(\frac{1}{2}\) from both sides:
\[
\frac{x}{3} \geq \frac{1}{4} - \frac{2}{4} \implies \frac{x}{3} \geq -\frac{1}{4}
\]

2. Multiply by 3:
\[
x \geq -\frac{3}{4}
\]

### e) \(3 < \frac{2x - 2}{8}\)

1. Multiply both sides by 8:
\[
24 < 2x - 2
\]

2. Add 2 to both sides:
\[
26 < 2x
\]

3. Divide by 2:
\[
13 < x \implies x > 13
\]

### f) \(\frac{2x - 3}{3} \leq \frac{3x - 2}{4}\)

1. Multiply both sides by 12 (the least common multiple of 3 and 4):
\[
4(2x - 3) \leq 3(3x - 2)
\]

2. Distributing:
\[
8x - 12 \leq 9x - 6
\]

3. Rearranging:
\[
-12 + 6 \leq 9x - 8x \implies -6 \leq x \implies x \geq -6
\]

### Summary of Solutions:
- a) \(x > \frac{5}{2}\)
- b) \(x > 4\)
- c) \(x > -50\)
- d) \(x \geq -\frac{3}{4}\)
- e) \(x > 13\)
- f) \(x \geq -6\)