Let's begin by graphing each inequality. Then, we will identify the overlapping region and find the coordinates of the vertices.
Inequality I
Let's graph the first inequality. By exchanging the inequality symbol to an equals sign, we can find the boundary line.
Inequality:& y≥- x-8
Boundary Line:& y=- 1x-8
The y-intercept of this boundary line is - 8 and the slope is - 1. The line will be solid as the inequality is not strict. Let's test the point ( 0, 0).
Since the point satisfies the inequality, we will shade the region that contains the point.
Inequality II
Let's graph the second inequality. We will determine the boundary line by exchanging the inequality to an equals sign. Then we should isolate y so that we may graph the line using the slope-intercept form.
2y=3x-20 ⇔ y=3/2x-10
Next, to determine which region to shade, we will test the point ( 0, 0).
Once more, the point satisfied the inequality so we will shade the region that contains the point.
Inequality III
We will follow the same process for the third inequality. Let's determine the boundary line and write it in slope-intercept form.
4y+x=24 ⇔ y=-1/4x+6
Now, we will test the point ( 0, 0), like before.
Let's draw the graphs of the inequalities on the same coordinate plane and highlight the vertices.
To determine the vertices we have to find the points of intersection of the boundary lines. Let's calculate the vertex formed by y=- x-8 and y=4x+22. To do this, we have to solve the system of equations related to these lines.
