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Determine the region subject to the following constraints:

[tex]\[

\begin{array}{c}

x + y \leq 20000 \\

x \geq 3000 \\

y \geq 4000

\end{array}

\][/tex]

Answer :

To determine the vertices of the feasible region defined by the constraints

[tex]$$
x + y \le 20000,\quad x \ge 3000,\quad y \ge 4000,
$$[/tex]

we analyze the intersections of the boundary lines.

1. First, consider the line [tex]$$x + y = 20000.$$[/tex]
When we set [tex]$$x = 3000,$$[/tex] the equation becomes
[tex]$$
3000 + y = 20000 \quad \Longrightarrow \quad y = 20000 - 3000 = 17000.
$$[/tex]
Thus, one vertex is
[tex]$$
(3000,\, 17000).
$$[/tex]

2. Next, again take the line [tex]$$x + y = 20000.$$[/tex]
Now set [tex]$$y = 4000,$$[/tex] so that
[tex]$$
x + 4000 = 20000 \quad \Longrightarrow \quad x = 20000 - 4000 = 16000.
$$[/tex]
This gives a second vertex:
[tex]$$
(16000,\, 4000).
$$[/tex]

3. Finally, consider the intersection of the vertical and horizontal constraints.
With [tex]$$x = 3000$$[/tex] and [tex]$$y = 4000,$$[/tex] we immediately have the vertex
[tex]$$
(3000,\, 4000).
$$[/tex]

These three points

[tex]$$
(3000,\, 17000),\quad (16000,\, 4000),\quad (3000,\, 4000)
$$[/tex]

are the extreme points of the feasible region defined by the constraints.

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