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**Topic**

Mr. Trung plans to invest 400 million dong in two accounts X and Y. In order to make a profit, item X must invest at least 100 million dong and the investment amount for item Y is not less than the amount for item X. Write a system of first-order inequalities with two unknowns to describe the two investments and represent the solution domain of the system of inequalities just found.

**Solution method – See details**

– Let x, y be the amount of money Trung invested in X and Y ., respectively

– Represent two investments in terms of x, y

– Use the given data to create a system of hidden inequalities x, y

– Determine the domain of the solution of the inequality on the coordinate plane

**Detailed explanation**

Let x, y (million VND) be the amount of money Mr. Trung invested in X and Y respectively. (\(x,y \ge 0\))

Because Trung invested 400 million dong in two accounts X and Y, we have \(x + y \le 400\)

To make a profit, X must invest at least 100 million dong, so we have \(x \ge 100\).

The investment amount for Y is not less than the amount for X, so we have \(y \ge x\) nice \(x – y \le 0\)

Then we have the system of inequalities: \(\left\{ {\begin{array}{*{20}{c}}{x + y \le 400}\\{x \ge 100}\\{x – y \le 0}\end{ array}} \right.\)

We draw four lines:

\({d_1}\): x + y = 400 is a straight line passing through two points with coordinates (400;0) and (0;400);

\({d_2}\): x = 100 is a line parallel to the Oy axis and passing through the point with coordinates (100, 0);

\({d_3}\): x – y = 0 is a straight line passing through two points with coordinates (0,0) and (1;1).

We define each solution domain of each inequality in the system, crossing out the parts that are not in the solution domain of each inequality.

The solution domain of the system of inequalities is the region in quadrilateral ABC with the following figure:

Where A(100;100), B(100,300) and C(200;200)

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