We are given a graph and asked to find and correct the mistake. Let's start by looking at the given graph.
Let's take a look at the different individually.
⎩⎪⎪⎨⎪⎪⎧x≥-2y>221x+y≥0(I)(II)(III)
In order to find the error the student has made, we will graph each inequality. Let's start with Inequality (I).
Inequality (I)
To graph the inequality, we have to draw the . The equation of a boundary line is written by replacing the inequality symbol from the inequality with an equals sign.
Inequalityx ≥ -2Boundary Linex = -2
Notice that the equation for this boundary line only has one . This equation tells us that each and every point that lies on the line will have an
x-coordinate equal to
-2. This gives us a . Also, the inequality is , so the points on the boundary line
are included in the . We show this by drawing a solid line.
The inequality x≥-2 describes all values of x that are greater than or equal to -2. This means that every possible (x,y) coordinate pair with an x-value that is greater than or equal to -2 needs to be included in the shading.
Inequality (II)
To graph the inequality, we will first draw the boundary line. As before, we will write the equation of the boundary line by replacing the inequality symbol from the inequality with an equals sign.
Inequalityy > 2Boundary Liney = 2
Notice that the equation for this boundary line only has one variable. This equation tells us that each and every point that lies on the line will have a
y-coordinate equal to
2. This gives us a . Also, the inequality is , so the points on the boundary line are
not included in the solution set. We show this by drawing a dashed line.
The inequality y>2 describes all values of y that are greater than 2. This means that every possible (x,y) coordinate pair with a y-value that is greater than 2 needs to be included in the shading.
As we can see, we have shaded the opposite region to the one the student has shaded in their solution. It means the student shaded the solution set of this inequality incorrectly.
Inequality (III)
In order to graph the Inequality (III), we will first isolate the
y-variable to rewrite the inequality in .
Let's write the equation of the boundary line of this inequality, as we have done for the two previous inequalities.
Inequalityy ≥ -21xBoundary Liney = -21x
Now that the boundary line is in slope-intercept form, we can determine its
m and
b.
Slope-Intercept Form: Boundary Line: y=mx+by=-21x+0
We can use the slope and the
y-intercept to draw the boundary line. Note that the boundary line will be solid, because the inequality is .
Note that this graph does not match the student's, which means
the student has drawn the graph of this inequality's boundary line incorrectly. By testing a point that is not on the boundary line we can determine which region we should shade. Let's test the point
(1,0). If it satisfies the inequality, we will shade the region that contains the point. Otherwise, we will shade the opposite region.
y≥-21x
0≥?-21(1)
0≥-21 ✓
Since the test point
satisfies the inequality, we will shade the region that
contains the point.
Conclusion
Let's draw all the graphs and shaded regions in the same .
The correct solution set is the of all the shaded regions.