$\begin{array}{l}y>x+2 \\ y \leq-2 x-1\end{array}$

Step 1 ：We are given a system of inequalities: \(y > x + 2\) and \(y \leq -2x - 1\).

Step 2 ：The first inequality, \(y > x + 2\), represents a region above the line \(y = x + 2\).

Step 3 ：The second inequality, \(y \leq -2x - 1\), represents a region below the line \(y = -2x - 1\).

Step 4 ：The solution to the system of inequalities is the intersection of these two regions.

Step 5 ：We can plot these two lines and their corresponding regions to visualize the solution.

Step 6 ：The plot shows two lines and two shaded regions. The blue region represents the solutions to the inequality \(y > x + 2\), and the red region represents the solutions to the inequality \(y \leq -2x - 1\). The intersection of these two regions, which is the purple region, represents the solutions to the system of inequalities.

Step 7 ：However, the plot does not provide exact values for the solutions. To find the exact solutions, we need to solve the system of inequalities algebraically.

Step 8 ：The system of inequalities can be rewritten as: \[\begin{align*} y - x &> 2 \\ y + 2x &\leq -1 \end{align*}\]

Step 9 ：We can solve this system of inequalities by substitution or elimination. However, because the inequalities are not equalities, we need to be careful about the direction of the inequalities when we combine them.

Step 10 ：The solution to the system of inequalities is a region of the coordinate plane, not a single point.

Step 11 ：The point \((-1, 1)\) is the intersection of the lines \(y = x + 2\) and \(y = -2x - 1\). This point is the boundary between the regions that satisfy the inequalities.

Step 12 ：The region that satisfies the inequality \(y > x + 2\) is the region above the line \(y = x + 2\), and the region that satisfies the inequality \(y \leq -2x - 1\) is the region below the line \(y = -2x - 1\).

Step 13 ：Therefore, the region that satisfies both inequalities is the region above the line \(y = x + 2\) and below the line \(y = -2x - 1\). This region is bounded by the lines \(y = x + 2\), \(y = -2x - 1\), and the vertical line \(x = -1\).

Step 14 ：\(\boxed{\text{The solution to the system of inequalities is the region of the coordinate plane that is above the line } y = x + 2, \text{ below the line } y = -2x - 1, \text{ and to the left of the line } x = -1. \text{ This region is bounded by the lines } y = x + 2, y = -2x - 1, \text{ and } x = -1.}\)