Step 1:

The first inequality is $2x+y \geq 8$

Consider the equation $2x +y =8$

The points $(4,0) ,(0,8)$ satisfy the equation.

The graph of this is drawn as shown.

The line divides the xy plane into two half planes .

Consider the point $(0,0)$

We see $2(0) +0 \geq 8$

$0 \geq 8$ is false

Thus the inequality $2x+8 \geq 8$ represents the region above the line not containing the point (0,0) (including the line 2x+y=8)

Step 2:

The second inequality is $x+2y \leq 10$ ---------(2)

Consider the equation : $x+2y =10$

The points $(10,0) $ and $(0,5)$ satisfy the equation.

The graph of this line is drawn as shown.

The line divides the xy -plane into two half planes

Consider the point (0,0)

$0+2(0) \geq 10$ is false

Thus the inequality $x+2y \geq 10$ represents the region above the line not containing the point (0,0).

including the line $x+2y=10$

Step 3:

Hence the solution of the given system of linear inequalities is represented by the common shaded region, including the points on the lines.