To solve the equation involving the reciprocal of \(\frac{2}{3}(x - 4) = -10\), we first recognize that taking the reciprocal of a value means finding \(\frac{1}{\text{value}}\).
However, in this case, it seems like you want to isolate \(x\) in the given equation. To proceed, let's first rewrite the equation for clarity:
\[ \frac{2}{3}(x - 4) = -10 \]
Now, let's solve for \(x\).
- Multiply both sides by \(\frac{3}{2}\) to eliminate the fraction:
\[ x - 4 = -10 \cdot \frac{3}{2} \]
- Calculate \(-10 \cdot \frac{3}{2}\):
\[ -10 \cdot \frac{3}{2} = -15 \]
So we have:
\[ x - 4 = -15 \]
- Add 4 to both sides:
\[ x = -15 + 4 \]
\[ x = -11 \]
Thus, the solution to the equation \(\frac{2}{3}(x - 4) = -10\) is:
\[ \boxed{-11} \]