To solve the equation \(-\frac{2}{3}x + 9 = \frac{4}{3}x - 3\), we can follow these steps:
-
Move all terms involving \(x\) to one side and constants to the other side.
Start by adding \(\frac{2}{3}x\) to both sides:
\[ 9 = \frac{4}{3}x + \frac{2}{3}x - 3 \]
This simplifies to:
\[ 9 = \left(\frac{4}{3} + \frac{2}{3}\right)x - 3 \]
Combining the fractions on the right:
\[ 9 = \frac{6}{3}x - 3 \]
This simplifies to:
\[ 9 = 2x - 3 \]
-
Isolate \(2x\) by adding \(3\) to both sides:
\[ 9 + 3 = 2x \]
This simplifies to:
\[ 12 = 2x \]
-
Solve for \(x\) by dividing both sides by \(2\):
\[ x = \frac{12}{2} = 6 \]
So, the value of \(x\) is \(\boxed{6}\).