Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.

To solve the given system of inequalities graphically, we need to plot each inequality on a Cartesian plane.

Step 1: Plot the inequalities

1. Inequality 1: \(y \leq x - 2\)

   • First, plot the line \(y = x - 2\). This line has a slope of 1 and a y-intercept of -2.
   • The line divides the plane into two regions. Since the inequality is \(y \leq x - 2\), we shade the region below this line.

2. Inequality 2: \(y \leq -\frac{1}{2}x - 5\)

   • Plot the line \(y = -\frac{1}{2}x - 5\). This line has a slope of -1/2 and a y-intercept of -5.
   • Again, since the inequality is \(y \leq -\frac{1}{2}x - 5\), shade the region below this line as well.

Step 2: Identify the solution set

The solution set is the region where the shaded areas from both inequalities overlap.

Step 3: Find an intersection point (optional for graphical solution)

To find a point in the solution set, we can look for points of intersection between the two lines.

1. Set the equations equal to find their intersection: \[ x - 2 = -\frac{1}{2}x - 5 \] \[ 2x - 4 = -x - 10 \implies 3x = -6 \implies x = -2 \] \[ y = -2 - 2 = -4 \] Thus, the intersection point is \((-2, -4)\).

Conclusion

The intersection point \((-2, -4)\) lies in the solution set since it satisfies both inequalities.

Final Answer: The coordinates of a point in the solution set are \((-2, -4)\).