Short Answer 
To solve the inequalities, first graph each on the same coordinate system, using a dashed line for (x – y < -1) and a solid line for (3x + 5y leq 10). Next, find the intercepts of both equations to plot accurately, and finally, identify the intersecting shaded area which represents the solution region, confirming the correct option as D.
Step 1: Graph the Inequalities
Begin by graphing each inequality individually on the same coordinate system. The inequality x Рy < -1 should be represented as a dashed line because the points on the line itself are not included in the solution set. Conversely, the inequality 3x + 5y ≤ 10 is represented with a solid line, indicating that points on this line are included in the solution set.
Step 2: Determine Intercepts
Next, find the intercepts for both equations, which are crucial for accurately plotting the lines. For the equation x – y = -1, the y-intercept is found by setting x = 0 to get the point (0, 1), and the x-intercept is found by setting y = 0 to get the point (-1, 0). For the equation 3x + 5y = 10, set x = 0 to find the y-intercept (0, 2) and y = 0 for the x-intercept (3.33, 0).
Step 3: Identify the Solution Region
After plotting both lines, determine the solution region which represents all possible solutions that satisfy both inequalities. This region is found where the areas of shading from both graphs intersect. The valid solution region is highlighted as the darkest shaded area, indicating where both conditions are met. In this case, the correct option from a provided list is D.