1. **State the problem:** Find the angle between the x-axis and the line joining the points $(3, -1)$ and $(4, -2)$. 2. **Formula used:** The angle $\theta$ between the x-axis and a line through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$\theta = \tan^{-1}\left(\frac{y_2 - y_1}{x_2 - x_1}\right)$$ 3. **Calculate the slope:** $$m = \frac{-2 - (-1)}{4 - 3} = \frac{-2 + 1}{1} = \frac{-1}{1} = -1$$ 4. **Find the angle:** $$\theta = \tan^{-1}(-1)$$ 5. **Evaluate the inverse tangent:** The angle whose tangent is $-1$ is $-45^\circ$ or in radians $-\frac{\pi}{4}$. 6. **Interpretation:** Since the angle is measured from the positive x-axis, the line makes an angle of $45^\circ$ below the x-axis. **Final answer:** $$\boxed{45^\circ}$$