1. **State the problem:** We have a right triangle XYZ with hypotenuse $XY = 5$ and leg $YZ = 3$. We need to find the tangent of angle $X$. 2. **Recall the tangent formula:** For an angle in a right triangle, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. 3. **Identify sides relative to angle $X$:** - Opposite side to angle $X$ is $YZ = 3$. - Adjacent side to angle $X$ is $XZ$ (unknown). 4. **Find the length of $XZ$ using the Pythagorean theorem:** $$XY^2 = XZ^2 + YZ^2$$ $$5^2 = XZ^2 + 3^2$$ $$25 = XZ^2 + 9$$ $$XZ^2 = 25 - 9 = 16$$ $$XZ = \sqrt{16} = 4$$ 5. **Calculate $\tan(X)$:** $$\tan(X) = \frac{\text{opposite}}{\text{adjacent}} = \frac{YZ}{XZ} = \frac{3}{4}$$ **Final answer:** $\boxed{\frac{3}{4}}$