1. State the problem: Find the obtuse angle that the line segment AB forms with the x-axis, where A = $(4,-3)$ and B = $(8,5)$.\n2. Calculate the slope of line AB using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.\n\n$m = \frac{5 - (-3)}{8 - 4} = \frac{8}{4} = 2$.\n3. The angle $\theta$ the line makes with the x-axis is given by $\theta = \arctan(m)$.\n\n$\theta = \arctan(2)$.\n4. Calculate $\theta$ in degrees: $\theta = \arctan(2) \approx 63.435^\circ$.\n5. Since we want the obtuse angle between the line and the x-axis, it is the supplement of $63.435^\circ$: $180^\circ - 63.435^\circ = 116.565^\circ$.\n\nFinal Answer: The obtuse angle line AB makes with the x-axis is approximately $116.565^\circ$.