1. **State the problem:** We have a right triangle ABC with a right angle at A. The other two angles are given as $(3x + 9)^\circ$ and $(4x - 3)^\circ$. We need to find the value of $x$. 2. **Recall the rule:** The sum of the angles in any triangle is $180^\circ$. 3. **Set up the equation:** Since angle A is $90^\circ$, the sum of angles B and C must be $90^\circ$: $$ (3x + 9) + (4x - 3) = 90 $$ 4. **Simplify the equation:** $$ 3x + 9 + 4x - 3 = 90 $$ $$ 7x + 6 = 90 $$ 5. **Isolate $x$:** $$ 7x + 6 = 90 $$ $$ 7x = 90 - 6 $$ $$ 7x = 84 $$ 6. **Divide both sides by 7:** $$ \cancel{7}x = \frac{84}{\cancel{7}} $$ $$ x = 12 $$ 7. **Final answer:** $$ \boxed{12} $$ This means the value of $x$ is 12.