Let's start with the equation:

\[
- \frac{3}{5}(x - 10) = \frac{4}{7} \left(14 - \frac{1}{8}\right)
\]

First, let's simplify the right side:

\[
14 - \frac{1}{8} = \frac{112}{8} - \frac{1}{8} = \frac{111}{8}
\]

Now substitute this back into the equation:

\[
- \frac{3}{5}(x - 10) = \frac{4}{7} \cdot \frac{111}{8}
\]

Now calculate the right side:

\[
\frac{4}{7} \cdot \frac{111}{8} = \frac{4 \times 111}{7 \times 8} = \frac{444}{56}
\]

We can simplify \(\frac{444}{56}\) by dividing the numerator and the denominator by 4:

\[
\frac{444 \div 4}{56 \div 4} = \frac{111}{14}
\]

So the equation now looks like this:

\[
- \frac{3}{5}(x - 10) = \frac{111}{14}
\]

Next, let's get rid of the negative sign by multiplying both sides by -1:

\[
\frac{3}{5}(x - 10) = -\frac{111}{14}
\]

Now, to eliminate the fraction on the left side, we can multiply both sides by \(\frac{5}{3}\):

\[
x - 10 = -\frac{111}{14} \cdot \frac{5}{3}
\]

Calculating the right side:

\[
-\frac{111 \cdot 5}{14 \cdot 3} = -\frac{555}{42}
\]

So we have:

\[
x - 10 = -\frac{555}{42}
\]

Next, we add 10 to both sides. To add 10, we express it with a denominator of 42:

\[
10 = \frac{420}{42}
\]

Therefore, we have:

\[
x = -\frac{555}{42} + \frac{420}{42} = \frac{-555 + 420}{42} = \frac{-135}{42}
\]

We can simplify \(-\frac{135}{42}\) by dividing both the numerator and the denominator by 3:

\[
x = -\frac{45}{14}
\]

Thus, the solution is:

\[
\boxed{-\frac{45}{14}}
\]