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Question
Show that the solution set of the following linear in equations is an unbounded set:
x + y ≥ 9
3x + y ≥ 12
x ≥ 0, y ≥ 0
Solution
Converting the inequations to equations, we obtain:
x + y = 9, 3x + y = 12 , x = 0, y = 0
x + y = 9: This line meets the x-axis at (9, 0) and y-axis at (0, 9). Draw a thick line through these points.
Now, we see that the origin (0, 0) does not satisfy the inequation x + y\[\geq\]9 Therefore, the potion that does not contain the origin is the solution set to the inequaltion x + y\[\geq\]9 3x + y = 12: This line meets the x-axis at (4, 0) and y-axis at (0, 12). Draw a thick line through these points.
Now, we see that the origin (0, 0) does not satisfy the inequation 3x + y\[\geq\]12. Therefore, the potion that does not contain the origin is the solution set to the inequaltion 3x + y\[\geq\]
Also, x ≥ 0, y ≥ 0 represents the first quadrant. Hence, the solution set lies in the first quadrant.
We see that in this solution set, the shaded region is unbounded (infinite). Hence, the solution set to the given set of inequalities is an unbounded set.
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