Topic
The polygon domain ABCD in Figure 9 is the solution domain of the system of inequalities:
A. \(\left\{ {\begin{array}{*{20}{c}}{x + y \le 4}\\{x + y \ge – 1}\\{x – y \le 2}\\{x – y \ge – 2}\end{array}} \right.\) B. \(\left\{ {\begin{array}{*{20}{c}}{x – y \le 4}\\{x – y \ge – 1}\\{x + y \le 2}\\{x + y \ge – 2}\end{array}} \right.\) 

C. \(\left\{ {\begin{array}{*{20}{c}}{x + y \le 1}\\{x + y \ge – 4}\\{x – y \le 2}\\{x – y \ge – 2}\end{array}} \right.\) D. \(\left\{ {\begin{array}{*{20}{c}}{x – y \le 1}\\{x – y \ge – 4}\\{x + y \le 2}\\{x + y \ge – 2}\end{array}} \right.\) 
Solution method – See details
 Step 1: Determine the equation of the line dividing the plane into two parts of the form \(ax + by = c\)
 Step 2: Take a point \(M\left( {{x_o};{y_o}} \right)\) in the solution domain of the inequality, replace the coordinates of the point M into \(ax + by\) and compare Compare with c to determine the required inequality
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Detailed explanation
Choose A
+) Call d_{first} is a straight line passing through two points A and D. The line intersects the two coordinate axes at two points (– 2; 0) and (0; 2) so the equation for the line d is: \(\frac{x}{{ – 2}} + \frac{y}{2} = 1 \Leftrightarrow x – y = – 2\)
Taking the point \(O\left( {0;0} \right)\) we have \(0 – 0 = 0 > – 2\)
Since the point O belongs to the solution domain of the system of inequalities, we have the inequality \(x – y \ge – 2\)
+) Let \({d_2}\) be a straight line passing through two points A and D. The line intersects the two coordinate axes at two points \(\left( {4;0} \right)\) and \(\ left( {0;4} \right)\)so the equation for the line d is: \(\frac{x}{4} + \frac{y}{4} = 1 \Leftrightarrow x + y = 4\)
Taking the point \(O\left( {0;0} \right)\) we have \(0 + 0 = 0 < 4\)
Since the point O belongs to the solution domain of the system of inequalities, we have the inequality \(x + y \le 4\)
+) Call d_{3} is a line passing through two points B and C. The line intersects the two coordinate axes at two points (2; 0) and (0; – 2), so the equation for the line d is: \(\frac{x}{2 } + \frac{y}{{ – 2}} = 1 \Leftrightarrow x – y = 2\)
Taking the point \(O\left( {0;0} \right)\) we have \(0 – 0 = 0 < 2\)
Since the point O belongs to the solution domain of the system of inequalities, we have the inequality \(x – y \le 2\)
Call d_{4} is a line passing through two points D and C. The line intersects the two coordinate axes at two points (– 1; 0) and (0; – 1) so the equation for the line d is: \(\frac{x}{ { – 1}} + \frac{y}{{ – 1}} = 1 \Leftrightarrow x + y = – 1\)
Take the point \(O\left( {0;0} \right)\) we have 0 + 0 =0 > 1
Since point O belongs to the domain of solutions of the system of inequalities, we have the inequality \(x + y \ge – 1\)
From this we have the following system of inequalities: \(\left\{ {\begin{array}{*{20}{c}}{x – y \ge – 2}\\{x + y \le 4} \\{x – y \le 2}\\{x + y \ge – 1}\end{array}} \right.\)
Choose A