$\left\{ \begin{array}{l} y=2 \text{ (red line)} \\ x= 6 \text{ (blue line)}\\ y=3x+2 \text{ (green line)} \\ y= -x+10 \text{ (orange line)}. \end{array}\right.$

2. Determine which bounded part of the plane you should select:

$y\ge 2$ means that you should take points with y-coordinates greater than or equal to 2 (top part of the coordinate plane that was formed by the red line);
$x\le 6$ means that you should take points with x-coordinates less than or equal to 6 (left part of the coordinate plane that was formed by the blue line);
for $y\le 3x+2$ you can check where the origin is placed. Since $0\le 3\cdot 0+2$ , the origin belongs to the needed part and you have to take the right part of the coordinate plane that was formed by green line.
for $y\le -x+10$ you can check where the origin is placed. Since $0\le -1\cdot 0+10$ , the origin belongs to the needed part and you have to take the bottom part of the coordinate plane that was formed by orange line.

3. According to the previous explanations, the shaded region is as in A diagram.

Answer: correct choice is A.

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