Graph the solution set of each system of inequalities.
$y>x^2+2$
$y\le x-2$

Verified step by step guidance

1. Step 1: Begin by graphing the boundary of the first inequality, $y>x^2+2$. The boundary is the parabola $y=x^2+2$. Since the inequality is 'greater than', use a dashed line to indicate that points on the line are not included in the solution set.
2. Step 2: Determine which side of the parabola $y=x^2+2$ to shade. Choose a test point not on the parabola, such as (0,0). Substitute into the inequality: $0>0^2+2$ which simplifies to $0>2$. This is false, so shade the region above the parabola.
3. Step 3: Next, graph the boundary of the second inequality, $y\le x-2$. This is a straight line with a slope of 1 and a y-intercept of -2. Use a solid line because the inequality is 'less than or equal to', indicating that points on the line are included in the solution set.
4. Step 4: Determine which side of the line $y=x-2$ to shade. Use the test point (0,0) again. Substitute into the inequality: $0\le 0-2$ which simplifies to $0\le -2$. This is false, so shade the region below the line.
5. Step 5: The solution set of the system of inequalities is the region where the shaded areas from both inequalities overlap. This is the area above the parabola $y=x^2+2$ and below the line $y=x-2$.