## Solve Lesson 13 Page 30 Math 10 SBT – Kite>

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 dfirst 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 d3 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 d4 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