$\"\"$

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The objective function is $\"\"$ .

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The constraints are

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$\"\"$

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Graph :

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Graph the inequalities and shade the required region.

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$\"\"$

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Note : The shaded region is the set of solution points for the objective function.

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The point $\"\"$ is not considered as $\"\"$ .

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Observe the graph:

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Tabulate the solutions of each of two system of inequalities and obtain the intersection points.

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System of boundary

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equations

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$\"\"$

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$\"\"$

\

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$\"\"$

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$\"\"$

\

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Solution (vertex points)

\"\"

\"\"

\"\"

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$\"\"$

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$\"\"$

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$\"\"$

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Find the value of objective function at the solution points for which it is minimum.

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At point $\"\"$ , $\"\"$ .

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At point $\"\"$ , $\"\"$ .

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At point $\"\"$ , $\"\"$ .

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At point $\"\"$ , $\"\"$ .

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At point $\"\"$ , $\"\"$ .

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Observe the values of $\"\"$ :

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The minimum value of $\"\"$ is $\"\"$ when $\"\"$ and $\"\"$ .

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$\"\"$

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The minimum value of $\"\"$ is $\"\"$ at $\"\"$ .