To solve the equation \( \frac{4x}{3} = -6 \), follow these steps:

1. **Isolate \( x \)**: Start by getting rid of the fraction. Multiply both sides of the equation by 3 to eliminate the denominator.

\[
3 \cdot \frac{4x}{3} = 3 \cdot (-6)
\]

This simplifies to:

\[
4x = -18
\]

2. **Solve for \( x \)**: Next, divide both sides by 4 to isolate \( x \).

\[
x = \frac{-18}{4}
\]

3. **Simplify the fraction**: Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

\[
x = \frac{-18 \div 2}{4 \div 2} = \frac{-9}{2}
\]

So, the solution to the equation \( \frac{4x}{3} = -6 \) is:

\[
x = -\frac{9}{2}
\]

Or in decimal form:

\[
x = -4.5
\]