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Question
Show that the solution set of the following system of linear inequalities is an unbounded region:
\[2x + y \geq 8, x + 2y \geq 10, x \geq 0, y \geq 0\]
Solution
\[\text{ We have }, \]
\[2x + y \geq 8 . . . . . \left( i \right)\]
\[x + 2y \geq 10 . . . . . \left( ii \right)\]
\[x \geq 0 . . . . . \left( iii \right)\]
\[y \geq 0 . . . . . \left( iv \right)\]
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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