Exercise 18 Page 161

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.

Inequality I

To graph a compound inequality, we can separate it into two cases. To make things a bit more simple, we will draw the graph of each case and then combine them.

Case I

The inequality $25\le x$ describes all values of $x$ that are greater than or equal to $25.$ This means that every coordinate pair with an $x\text{-}$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 $x\text{-}$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.

Inequality II

The inequality $y\le 110$ tells us that all coordinate pairs with a $y\text{-}$coordinate that is less than or equal to $110$ will be in the solution set of the inequality. Because the inequality is non-strict, the boundary line will be solid.

Inequality III

To write the equation of the boundary line for the third inequality, we will first isolate the $y\text{-}$variable. We can write the equation of the boundary line by changing the inequality symbol to an equals sign. Since this equation is in slope-intercept form, we can determine its slope $m$ and $y\text{-}$intercept $b$ to draw the line. Now that we know the slope and $y\text{-}$intercept, let's use these to draw the boundary line. Notice that the inequality is non-strict, which means that the line will be solid. To complete the graph, we will test a point that is not on the line and decide which region we should shade. Let's test the point $(0,0).$ If the point satisfies the inequality, we will shade the region that contains the point. Otherwise, we will shade the other region. Since the point does not satisfy the inequality, we will shade the region that does not contain the point.

Combining the Inequality Graphs

Let's draw the graphs of the inequalities on the same coordinate plane.

Now that we can see the overlapping region, let's highlight the vertices.

Looking at the graph, we can see that the line $y=110$ intersects the lines $x=25$ and $x=75$ at the points $(25,110)$ and $(75,110),$ respectively. These are two of the vertices. We know the $x\text{-}$coordinates of the other vertices are $25$ and $75.$ We will find their $y\text{-}$coordinates by substituting them into the equation $y=\text{-}\frac{4}{3}x+120.$ Let's do it! The third vertex is $(25,\frac{260}{3}).$ Next, we will determine the final vertex. The last vertex is $(75,20).$

Finding the Minimum Value

Substituting the vertices into the given equation, $C=8x+5y,$ we will determine its minimum value. Let's start with the vertex $(25,110).$ For the vertex $(25,110),$ the value of the equation is $750.$ We can determine the values of the equation for the other vertices in the same way.

Vertex	$C=8x+5y$	Value
$(25,110)$	$C=8(25)+5(110)$	$C=750$
$(75,110)$	$C=8(75)+5(110)$	$C=1150$
$\left(25,\frac{260}{3}\right)$	$C=8(25)+5\left(\frac{260}{3}\right)$	$C=633.\overline{3}$
$(75,20)$	$C=8(75)+5(20)$	$C=700$

Looking at the table, we can see that the minimum value of the equation is $C=633.\overline{3}.$