Our first step in finding the minimum value of the given equation is to graph the system and determine the vertices of the overlapping region. Substituting these vertices into the equation, we will find the minimum value.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 25 \leq x \leq 75 y \leq 100 8 x + 6 y \geq 720 (I) (II) (III)

Inequality I

To graph a compound inequality, we can separate it into two cases.

Compound Inequality : Case I : Case II : 25 \leq x \leq 75 25 \leq x \leq 75 25 \leq x \leq 75

To make things a bit more simple, we will draw the graph of each case and then combine them.

Case I

The inequality $25 \leq x$ describes all values of $x$ that are greater than or equal to $25 .$ This means that every coordinate pair with an $x -$ value that is greater than or equal to $25$ will be included in the shaded region. Notice that the inequality is not strict, so the boundary line will be solid.

Case II

Since the inequality represents the values of $x$ that are less than or equal to $75,$ every coordinate pair with an $x -$ value that is less than or equal to $75$ will be included in the shaded region. The boundary line will be solid because the inequality is non-strict.

Combining the Cases

Let's draw the graphs of both cases on the same coordinate plane and shade the overlapping region. This will give us the graph of the compound inequality.

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