Our first step in finding the minimum value of the given equation is to and determine the vertices of the overlapping region. Substituting these vertices into the equation, we will find the minimum value.
⎩⎪⎪⎨⎪⎪⎧25≤x≤75y≤1008x+6y≥720(I)(II)(III)
Inequality I
To graph a , we can separate it into two cases.
Compound Inequality: Case I: Case II: 25≤x≤75 25≤x≤75 25≤x≤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≤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 , so the 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.
Inequality II
The inequality y≤110 tells us that all coordinate pairs with a y-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-variable.
We can write the equation of the boundary line by changing the inequality symbol to an equals sign.
Inequality: Boundary Line: y≥-34x+120 y=-34x+120
Since this equation is in , we can determine its
m and
y-intercept
b to draw the line.
Slope-Intercept Form: Boundary Line: y=mx+b y=-34x+120
Now that we know the slope and
y-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.
y≥-34x+120
0≥?-34(0)+120
0≱120
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.
(25,110) and (75,110)
We know the
x-coordinates of the other vertices are
25 and
75. We will find their
y-coordinates by substituting them into the equation
y=-34x+120. Let's do it!
y=-34x+120
▼
Substitute 25 for x and simplify
y=-34(25)+120
y=-3100+120
y=3260
The third vertex is
(25,3260). Next, we will determine the final vertex.
y=-34x+120
▼
Substitute 75 for x and simplify
y=-34(75)+120
y=-3300+120
y=-100+120
y=20
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).
C=8x+5y
C=8(25)+5(110)
C=200+550
C=750
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
|
(25,3260)
|
C=8(25)+5(3260)
|
C=633.3
|
(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=633.3.