To solve the system of inequalities graphically, you need to plot each boundary line and then determine the region that satisfies both inequalities.

Inequality 1: $y > \frac{1}{2}x - 1$

1. Boundary Line: First, we consider the line $y = \frac{1}{2}x - 1$. This is a straight line with a slope of $\frac{1}{2}$ and a y-intercept of $-1$.
2. Graph: Plot the line by starting at the y-intercept (0, -1) and using the slope to find another point: from (0, -1), move up 1 unit and right 2 units to (2, 0). Since the inequality is strict ($>$), this line will be dashed.
3. Shading: Shade the region above the line because the inequality is $y > \frac{1}{2}x - 1$.

Inequality 2: $y \geq -\frac{3}{4}x - 6$

1. Boundary Line: Consider the line $y = -\frac{3}{4}x - 6$. This is a straight line with a slope of $-\frac{3}{4}$ and a y-intercept of $-6$.
2. Graph: Plot the line by starting at the y-intercept (0, -6) and using the slope to find another point: from (0, -6), move down 3 units and right 4 units to (4, -9). Since the inequality is inclusive ($\geq$), this line will be solid.
3. Shading: Shade the region above the line because the inequality is $y \geq -\frac{3}{4}x - 6$.

Solution Set

• The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap.
• Look for a point in this overlapping region.

Coordinate Example

Pick a point in the overlap. For example, let’s check the point (0, 0):

• For the first inequality: $0 > \frac{1}{2}(0) - 1$ simplifies to $0 > -1$, which is true.

• For the second inequality: $0 \geq -\frac{3}{4}(0) - 6$ simplifies to $0 \geq -6$, which is true.

Thus, (0, 0) is a point in the solution set.

To create the graph and verify your solution, make sure to plot both lines correctly and shade the appropriate regions. The point (0, 0) and any other point within the overlapping region should satisfy both inequalities.