b Let's graph the system of inequalities that we created in Part A. We can see that the first inequality involves only the x-variable while the second one involves both x and y-variables.
Graphing Inequality (I)
The equation of the boundary line is found by replacing the inequality symbol with an equals sign.
Inequality & Boundary Line
x ≤ 30 & x =30
We can see that this is a line whose x-intercept is 30. Additionally, since the inequality is not , the boundary line will be solid. The region represented by x ≤ 30 is all the points that are to the left of the boundary line.
Graphing Inequality (II)
We will find the boundary line in the same way for this inequality as well.
Inequality & Boundary Line
2x+2y ≤ 180 & 2x+2y = 180
The boundary line is not in form. Let's rewrite it.
2x+2y = 180
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Write in slope-intercept form
2y=180-2x
y=180-2x/2
y=180/2-2x/2
y=90-x
y=- x+90
From this last equation, we can see that the slope of the boundary line is - 1 and the y-intercept is 90. Also, since the initial inequality is not strict (≤), the line will be solid. Let's plot it in the previous coordinate plane.
Now, let's test the origin in the second inequality to determine which region we have to shade. If the point (0,0) produces a true statement, we will shade the region containing it. Otherwise, we will shade the other side of the line.
2x+2y ≤ 180
2( 0)+2( 0) ? ≤ 180
0 ≤ 180
Since 0 is less than or equal to 180, we've got a true statement. Hence, we will shade the region that contains the origin.
Final Graph
Finally, the graph that represents all possible solutions is the overlapping part. But, remember that x and y both represent the dimensions of the garden, so they have to be positive. Therefore, we will limit the region only to the first quadrant.
