Show that the solution set of the following system of linear inequalities is an unbounded region: $2 x + y \geq 8, x + 2 y \geq 10, x \geq 0, y \geq 0$ .

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Solution

$We have, 2 x + y \geq 8 . . . . . (i) x + 2 y \geq 10 . . . . . (ii) x \geq 0 . . . . . (iii) y \geq 0 . . . . . (iv)$

As, the solutions of the equation 2x + y = 8 are:

x 0 4 2

y 8 0 4

As, the solutions of the equation x + 2y = 10 are:

x 0 10 2

y 5 0 4

Now, the graph represented by the inequalities (i), (ii), (iii) and (iv) is as follows:

Since, the common shaded region is the solution set of the given set of inequalities.

So, the solution set of the given linear inequalities is an unbounded region.

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