To solve the equation

\[
-\frac{4}{7}x + \frac{7}{5} = \frac{1}{8} : \left(-1 + \frac{2}{3}\right),
\]

we first need to simplify the right side of the equation.

1. Simplify the expression \(-1 + \frac{2}{3}\):
\[
-1 + \frac{2}{3} = -\frac{3}{3} + \frac{2}{3} = -\frac{1}{3}.
\]

2. Now, replace the expression in the equation:
\[
-\frac{4}{7}x + \frac{7}{5} = \frac{1}{8} : -\frac{1}{3}.
\]

3. Now perform the division \(\frac{1}{8} : -\frac{1}{3}\):
\[
\frac{1}{8} \div -\frac{1}{3} = \frac{1}{8} \cdot -\frac{3}{1} = -\frac{3}{8}.
\]

Now we have the new equation:
\[
-\frac{4}{7}x + \frac{7}{5} = -\frac{3}{8}.
\]

Next, we solve for \(x\):

4. Isolate the term with \(x\):
\[
-\frac{4}{7}x = -\frac{3}{8} - \frac{7}{5}.
\]

To combine the two fractions on the right, we need a common denominator, which is 40 (the least common multiple of 8 and 5).

5. Convert \(-\frac{3}{8}\) to have a denominator of 40:
\[
-\frac{3}{8} = -\frac{3 \cdot 5}{8 \cdot 5} = -\frac{15}{40}.
\]

6. Convert \(-\frac{7}{5}\) to have a denominator of 40:
\[
-\frac{7}{5} = -\frac{7 \cdot 8}{5 \cdot 8} = -\frac{56}{40}.
\]

7. Now combine these fractions:
\[
-\frac{3}{8} - \frac{7}{5} = -\frac{15}{40} - \frac{56}{40} = -\frac{71}{40}.
\]

Now our equation is:
\[
-\frac{4}{7}x = -\frac{71}{40}.
\]

8. Now, multiply both sides by -1:
\[
\frac{4}{7}x = \frac{71}{40}.
\]

9. To isolate \(x\), multiply both sides by \(\frac{7}{4}\):
\[
x = \frac{71}{40} \cdot \frac{7}{4} = \frac{71 \cdot 7}{40 \cdot 4} = \frac{497}{160}.
\]

Thus, the solution for \(x\) is:
\[
\boxed{\frac{497}{160}}.
\]