To graph the inequalities, begin with the boundary lines. The inequality y<-x+7 has the boundary line y=-x+7. Since the inequality is strict, the line should be dashed and, in this case, the shaded region lies below the line.
Similarly, y<-0.5x+5 has the boundary line y=-0.5x+5. Since the inequality is not strict, the line is solid. In addition, the region to be shaded lies below the line. This inequality will be graphed on the same coordinate plane.
Notice that the individual solution sets overlap in a portion of the plane. This overlapping region is the solution set of the system. All the points in this region satisfy both inequalities simultaneously. In the next graph, only the common region is plotted.
Finally, since the boundary lines in their entirety are not part of the solution set, crop them only to show the edges of the overlapping region.
Write a system of inequalities that describes the shaded region.
The second inequality represent the region below the boundary line y=-1.5x−6. 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 $90 and instructs him to get enough food so that they can feed 10 people. If burritos cost $5 each and tacos cost $3 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 b and t.
Any point in this region corresponds to a combination on burritos and tacos that costs less than $90 and feeds at least 10 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 12 tacos and 8 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.