To solve the equation
\[ \frac{x + 8}{4} = \frac{2x - 9}{3}, \]
we can start by eliminating the fractions. We can do this by finding a common denominator, which in this case is 12. We can multiply both sides of the equation by 12:
\[ 12 \cdot \frac{x + 8}{4} = 12 \cdot \frac{2x - 9}{3}. \]
This simplifies to:
\[ 3(x + 8) = 4(2x - 9). \]
Now distribute on both sides:
\[ 3x + 24 = 8x - 36. \]
Next, we'll isolate \(x\). Start by getting all terms involving \(x\) on one side and constants on the other:
\[ 24 + 36 = 8x - 3x. \]
This simplifies to:
\[ 60 = 5x. \]
Now, divide by 5:
\[ x = 12. \]
To verify, we can substitute \(x = 12\) back into the original equation:
The left-hand side:
\[ \frac{12 + 8}{4} = \frac{20}{4} = 5. \]
The right-hand side:
\[ \frac{2(12) - 9}{3} = \frac{24 - 9}{3} = \frac{15}{3} = 5. \]
Both sides are equal, confirming that the solution is correct.
Thus, the solution is
\[ \boxed{12}. \]