Let's solve each equation step by step.

### Problem 1:
**Equation:**
\[
\frac{4}{7}x - x = -\frac{9}{14}
\]

**Step 1:** Rewrite \( x \) with a common denominator:
\[
x = \frac{7}{7}x
\]
So, the equation becomes:
\[
\frac{4}{7}x - \frac{7}{7}x = -\frac{9}{14}
\]
This simplifies to:
\[
\frac{4 - 7}{7}x = -\frac{9}{14}
\]
Which simplifies further to:
\[
-\frac{3}{7}x = -\frac{9}{14}
\]

**Step 2:** Multiply both sides by -1 to eliminate the negative sign:
\[
\frac{3}{7}x = \frac{9}{14}
\]

**Step 3:** Multiply both sides by \(\frac{7}{3}\) to isolate \(x\):
\[
x = \frac{9}{14} \cdot \frac{7}{3}
\]

**Step 4:** Simplify the right-hand side:
\[
x = \frac{9 \times 7}{14 \times 3} = \frac{63}{42}
\]

You can simplify \(\frac{63}{42}\) to:
\[
x = \frac{3}{2} \quad \text{(since } 63/42 = 3/2\text{)}
\]

### Problem 2:
**Equation:**
\[
\frac{1}{2}x + \frac{3}{5} : x = -\frac{2}{3}
\]

This can be interpreted as:
\[
\frac{1}{2}x + \frac{3}{5} \div x = -\frac{2}{3}
\]

**Step 1:** Rewrite the division as multiplication:
\[
\frac{1}{2}x + \frac{3}{5} \cdot \frac{1}{x} = -\frac{2}{3}
\]

**Step 2:** Multiply through by \( x \) (assuming \( x \neq 0 \)):
\[
\frac{1}{2}x^2 + \frac{3}{5} = -\frac{2}{3}x
\]

**Step 3:** Rearranging gives us a standard quadratic form:
\[
\frac{1}{2}x^2 + \frac{2}{3}x + \frac{3}{5} = 0
\]

**Step 4:** To eliminate the fractions, multiply through by the least common multiple of 2, 3, and 5, which is 30:
\[
30 \left(\frac{1}{2}x^2\right) + 30 \left(\frac{2}{3}x\right) + 30 \left(\frac{3}{5}\right) = 0
\]
This results in:
\[
15x^2 + 20x + 18 = 0
\]

**Step 5:** You can use the quadratic formula to solve \( ax^2 + bx + c = 0 \):
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \( a = 15, b = 20, c = 18 \).

Substituting values:
\[
x = \frac{-20 \pm \sqrt{20^2 - 4 \cdot 15 \cdot 18}}{2 \cdot 15}
\]
Calculating \( b^2 - 4ac \):
\[
400 - 1080 = -680
\]
Since the discriminant is negative, there are no real solutions for this equation. The solutions would be complex numbers.

### Summary of Results:
1. For the first equation:
\[
x = \frac{3}{2}
\]
2. For the second equation:
- No real solution (complex solutions exist).