The following system of inequalities contains two conditions on the variables and Systems of inequalities are often illustrated graphically in a coordinate plane, where the inequalities define a region.
To graph the inequalities, begin with the boundary lines. The inequality has the boundary line Since the inequality sign is the line should be dashed, and the region below the line is shaded.
Similarly, has the boundary line The inequality sign is so the line is dashed and the region below the line is shaded.
Notice that the individual solution sets overlap for a portion of the graph. This overlapping region is the solution set of the system. The points in this region are all the points that satisfy both inequalities. In this case, this is the purple region.
Lastly, since the boundary lines in their entirety are not part of the solution set, trim them to only show the borders of the overlapping region.
Write a system of inequalities that describes the shaded region.
To determine the system shown by the region, we can start with the boundary lines. The dashed line is vertical. If it were extended, we'd see that it passes through the -axis at Additionally, since the region to the right of is shaded, values of that are greater than are included. Since the line is dashed, is not included in the set. This means the inequality symbol is strict. Thus, The solid line is not vertical, so we can write its equation in slope-intercept form. Let's start by identifying two points on the line.
The second inequality represent the region below the boundary line The inequality isn't strict so the line should be solid.
The region that contains the solution set of both inequalities is shown by the overlapping red and blue shading.
The region is the same. Thus, the system of inequalities created is correct.
Marco's mother asks him to buy burritos and tacos from the restaurant near their house. She gives him and instructs him to get enough food so that they can feed people. If burritos cost each and tacos cost each, how many of each can he buy?
The purple region represents the solution set that satisfies both inequalities. Since Marco can't buy a negative number of burritos or tacos we're only interested in the positive values of and
Any point in this region corresponds to a combination on burritos and tacos that costs less than and feeds at least people. Let's look at the corners of this region.
The marked points represent minimum and maximum possibilities.
Since we don't know how hungry the guests are, or what their preferences are, Marco should buy both burritos and tacos. Let's choose a point in the middle of the region.
One possibility is that Marco can purchase tacos and burritos. That way, there will probably be enough food, and he'll have money left. Note that even though decimal numbers are a part of the solution set, the answer should be given in whole numbers, assuming you can't buy a part of a taco or burrito.