Let's solve each equation step-by-step.

### a) \( x - \frac{1}{15} = \frac{1}{10} \)

1. Add \(\frac{1}{15}\) to both sides:
\[
x = \frac{1}{10} + \frac{1}{15}
\]

2. To add the fractions, find a common denominator. The least common multiple of 10 and 15 is 30.
- Convert \(\frac{1}{10}\) to \(\frac{3}{30}\).
- Convert \(\frac{1}{15}\) to \(\frac{2}{30}\).

3. Now, you can add the fractions:
\[
x = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}
\]

### Solution to a)
\( x = \frac{1}{6} \)

---

### b) \( \frac{2}{3}x + \frac{1}{7} = \frac{5}{3} \)

1. Subtract \(\frac{1}{7}\) from both sides:
\[
\frac{2}{3}x = \frac{5}{3} - \frac{1}{7}
\]

2. To subtract the fractions, find a common denominator. The least common multiple of 3 and 7 is 21.
- Convert \(\frac{5}{3}\) to \(\frac{35}{21}\).
- Convert \(\frac{1}{7}\) to \(\frac{3}{21}\).

3. Now you can subtract the fractions:
\[
\frac{2}{3}x = \frac{35}{21} - \frac{3}{21} = \frac{32}{21}
\]

4. Multiply both sides by \(\frac{3}{2}\) to isolate \(x\):
\[
x = \frac{32}{21} \cdot \frac{3}{2} = \frac{32 \cdot 3}{21 \cdot 2} = \frac{96}{42} = \frac{16}{7} \quad \text{(after simplifying)}
\]

### Solution to b)
\( x = \frac{16}{7} \)

---

### c) \( \left( \frac{5}{6}x + 3 \right)^2 = \frac{4}{9} \)

1. Take the square root of both sides (consider both the positive and negative roots):
\[
\frac{5}{6}x + 3 = \frac{2}{3} \quad \text{or} \quad \frac{5}{6}x + 3 = -\frac{2}{3}
\]

#### First case:
- From \(\frac{5}{6}x + 3 = \frac{2}{3}\):
1. Subtract 3 from both sides:
\[
\frac{5}{6}x = \frac{2}{3} - 3 = \frac{2}{3} - \frac{9}{3} = -\frac{7}{3}
\]
2. Multiply both sides by \(\frac{6}{5}\):
\[
x = -\frac{7}{3} \cdot \frac{6}{5} = -\frac{42}{15} = -\frac{14}{5}
\]

#### Second case:
- From \(\frac{5}{6}x + 3 = -\frac{2}{3}\):
1. Subtract 3 from both sides:
\[
\frac{5}{6}x = -\frac{2}{3} - 3 = -\frac{2}{3} - \frac{9}{3} = -\frac{11}{3}
\]
2. Multiply both sides by \(\frac{6}{5}\):
\[
x = -\frac{11}{3} \cdot \frac{6}{5} = -\frac{66}{15} = -\frac{22}{5}
\]

### Solutions to c)
1. \( x = -\frac{14}{5} \)
2. \( x = -\frac{22}{5} \)

---

### Summary of Solutions:
- a) \( x = \frac{1}{6} \)
- b) \( x = \frac{16}{7} \)
- c) \( x = -\frac{14}{5} \) or \( x = -\frac{22}{5} \)