Graph the system of inequalities. Then find the coordinates of the vertices.
$$\begin{array}{l}y\ge -4\\ x\ge 6\end{array}$$

Step 1 ：The problem is asking to graph the system of inequalities and find the coordinates of the vertices. The system of inequalities consists of two inequalities: $y\ge -4$ and $x\ge 6$.

Step 2 ：The first inequality, $y\ge -4$, is a horizontal line that passes through the point (0, -4) and includes all the points above it.

Step 3 ：The second inequality, $x\ge 6$, is a vertical line that passes through the point (6, 0) and includes all the points to the right of it.

Step 4 ：The intersection of these two inequalities forms a region in the upper right quadrant of the coordinate plane. The vertices of this region are the points where the lines intersect the axes.

Step 5 ：To find the coordinates of the vertices, we need to find the points where the lines intersect the axes. The line $y=-4$ intersects the y-axis at (0, -4) and the line $x=6$ intersects the x-axis at (6, 0).

Step 6 ：The vertices of the region defined by the system of inequalities are at the points $\boxed{{\displaystyle (0,-4)}}$ and $\boxed{{\displaystyle (6,0)}}$.