Q:

which graph shows the solution to the system of inequalities? y<1/2x-2 y≤-2x+4

Accepted Solution

A:
Answer:First of all, we need to graph the equation of the two lines and find each region separately. The first line is:[tex]y=\frac{1}{2}x-2[/tex] This line has been plotted in the first figure below. To find the feasible region, let's take a random point and test the inequality, so let's take [tex](0,0)[/tex][tex]y<\frac{1}{2}x-2 \\ \\ 0<\frac{1}{2}(0)-2 \\ \\ 0<-2 \ False![/tex]Since for (0,0) the inequality is false, then the region for this inequality is not where the point (0, 0) lies, but the other region, that is, the one under the line as indicated in the second figure. Keep in mind that points on the border of this region, that is, points that lies on the line aren't included in the region because the symbol < tells us that the equality is not included.The second line is:[tex]y=-2x+4[/tex] This line has been plotted in the third figure below. To find the feasible region, let's take (0,0) again, so let's take [tex](0,0)[/tex][tex]y\leq -2x+4\\ \\ 0\leq -2(0)+4 \\ \\ 0<4 \ True![/tex]Since for the point (0,0) the inequality is true, then the region for this inequality is where this point lies as indicated in the fourth figure. Keep in mind that points on the border of this region, that is, points that lies on the line are included in the region because the symbol ≤ tells us that the equality is included._____________________Finally, the solution to the system of inequalities is the one where the red and blue region overlap as indicated in the fifth figure below.