We will first , then name the vertices of the overlapping region, which can be also called feasible. Note that two of the given inequalities contain only the
x-variable and two
− only the
y-variable.
⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧x≤5y≤4x≥0y≥0(I)(II)(III)(IV)
We will graph both cases, for
x- and for
y-values, separately. Next, we will combine the graphs and identify the overlapping region. Let's start!
Inequalities I and III
First, let's graph the inequalities which contain only the
x-variable.
Case I: Case II: x≤5 x≥0
Note that we can form a from these two cases.
Case I: Case II: Compound Inequality: 0≤x≤5 0≤x 0≤x≤5
This compound inequality describes all values of
x that are
greater than or equal to 0 and less than or equal to 5. This means that all with an
x-coordinate that is
greater than or equal to 0 and less than or equal to 5 will be in the solution set. Note that both inequalities are , so the boundary lines will be solid.
Inequalities II and IV
For inequalities containing only the
y-variable we will follow a similar process. Let's form the compound inequality.
Case I: Case II: Compound Inequality: 0≤y≤4 0≤y 0≤y≤4
This compound inequality describes all values of
y that are
greater than or equal to 0 and less than or equal to 4. This means that all coordinate pairs with a
y-coordinate that is
greater than or equal to 0 and less than or equal to 4 will be in the solution set. Note that both inequalities are non-strict, so the boundary lines will be solid.
Combining the Inequality Graphs
Let's draw the graphs of the inequalities on the same .
Now that we can see the overlapping region, let's highlight the vertices.
Looking at the graph, we can see that the line
y=0 intersects the lines
x=0 and
x=5 at the points
(0,0) and
(5,0), respectively. These are two of the vertices.
(0,0) and (5,0)
Similarly, we can see that the line
y=4 intersects the lines
x=0 and
x=5 at the points
(0,4) and
(5,4), respectively. We have identified two remaining vertices.
(0,4) and (5,4)