1. **Stating the problem:** We have a right triangle with legs labeled $3x$ and $4x$, and the hypotenuse labeled $3$. We need to find the value of $x$. 2. **Formula used:** In a right triangle, the Pythagorean theorem applies: $$a^2 + b^2 = c^2$$ where $a$ and $b$ are the legs and $c$ is the hypotenuse. 3. **Apply the formula:** Substitute $a = 3x$, $b = 4x$, and $c = 3$: $$ (3x)^2 + (4x)^2 = 3^2 $$ 4. **Simplify:** $$ 9x^2 + 16x^2 = 9 $$ $$ 25x^2 = 9 $$ 5. **Solve for $x^2$:** $$ x^2 = \frac{9}{25} $$ 6. **Find $x$:** $$ x = \pm \frac{3}{5} $$ 7. **Interpretation:** Since lengths are positive, we take the positive value: $$ x = \frac{3}{5} $$ **Final answer:** $x = \frac{3}{5}$