Graph the given system and determine the vertices of the overlapping region.
Graph:
Vertices: (0,0), (0,2), (4,0)
Practice makes perfect
Our first step in finding the vertices is to graph the system and determine the overlapping region.
x≥ 0 & (I) y≥ 0 & (II) x+2y<4 & (III)
Inequality I
The inequality x≥ 0 tells us that all coordinate pairs with an x-coordinate that is greater than or equal to 0 will be in the solution set of the inequality. Thus, the boundary line will be vertical. Note that the inequality is non-strict, which means that the line will be solid.
Inequality II
The inequality y≥ 0 tells us that all coordinate pairs with a y-coordinate that is greater than or equal to 0 will be in the solution set of the inequality. In this case the boundary line will be horizontal. It will also be solid, since the inequality is non-strict.
Inequality III
To graph the inequality, we will have to determine the boundary line first. We can do it by changing the inequality symbol to an equals sign.
Inequality:& x+2y< 4
Boundary Line:& x+2y=4
Let's rewrite this equation in slope-intercept form.
Since this equation is in slope-intercept form, we can determine its slope m and y-intercept b to draw the line.
Slope-Intercept Form:& y= mx+b
Boundary Line:& y= -1/2x+2
Now that we know the slope and y-intercept, let's use these to draw the boundary line. Notice that the inequality is strict, which means that the line will be dashed.
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.
