You approach this by identifying the line that bounds the shaded area and the side of the line the shaded area represents.
The line that has positive slope has a slope of 1 (it goes 1 square up for 1 square to the right), and it has a y-intercept of -2. Thus its equation is
y = x - 2
The shaded area is below this solid line, so corresponds to the solution to
y ≤ x - 2
Only one answer selection includes this inequality—the last one. You answer this by making the selection of that system of inequalities.
y ≤ x - 2
y ≤ -3x -6
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Had there been more choices that included the inequality we found, you would then look at the other line. It has a slope of -3 (down 3 squares for each square to the right) and it intersects the y-axis at y = -6. Thus its equation is
y = -3x - 6
Again, the shaded area is below the line (y values are less than those defined by the line), so the shaded area corresponds to the solution of
y ≤ -3x - 6
Now, you have both of the inequalities in the system of inequalities. The shaded area is the area that belongs to the solution sets of both of them.