Graph the solution set of each system of inequalities.

Question

$x\le4$

$x\ge0$

$y\ge0$

$x+2y\ge2$

Accepted Answer

Welcome back everyone. In this problem, we want to graph the solution set of X is less than or equal to seven, X is greater than, or equal to three Y greater than, or equal to one and X minus Y is less than, or equal to three. In our answer choices, we have possible solutions. So that's a possible solution in a possible one in B, then we have choice C and then D says there's no solution set. And below those, I've also put a graph here that can help us to easily graph our problem as we go along. OK? Now let's go ahead and see if we can try to solve by plotting each inequality on our graph. Let's start with the first one, X is less than sorry, X is less than, or equal to seven. No, the graph of or sorry. The in the graph of the inequality X is less than or equal to seven is going to show the graph of X equals seven with one region shaded. So first let's try to plot the graph of X equals seven and then figure out which region we're going to shade. Now that graph is basically going to be a vertical line that's passing to the 0.70. OK. So it's going to be this vertical line. And no, the question is, where do we share? Well, for one, we know that our vertical line is solid since the inequality symbol is less than our equal to. So it includes the values for seven. And then it's just telling us we want all the X values that are less than our equal to seven. So it's going to be everything to the left of the line. OK? So as a matter of fact, let's share that in yellow here. So that OK, it's gonna look something like this. Matter of fact, I think I need a bigger marker here. It's gonna be everything over this side. OK? So all of this represents the inequality X is less than or equal to seven. Now let's go ahead and try to solve for the other. The other one is X just put that in red is greater than, or equal to three. No, just like our problem before it's pretty similar. OK? Then we know that we go to the point where all of the X values are three, we plot the line X equals three. Oops that's two should actually be here. OK? So we go to the point where X is three and no, since it's X is greater than or equal to three, we're going to shade everything after the line or to the right of the line. So we're going to go in this direction. Ok. So our shade in red here represents X is greater than or equal to three. OK. And you know, let me just try to get it as close to the line as possible, but remember it's not supposed to go over the line. OK. So I can take out this section here a little bit. Let's just mix it back with yellow and red. All right, let's just shade it back yellow. OK? So this is going to be X is greater than our equal to three. Now, let's move on to our third. Inequality Y is greater than, or equal to one. OK? So let's put Y is greater than, or equal to one here and like the others, we know that if we go to the point where Y equals one, OK? Our line is going to pass through that point, OK? Or it's gonna pass through all the points where the Y values are equal to one and because it's greater than, or equal to our line will be solid, but include all of the or include the region rather where our Y coordinates. OK? Are greater than one. It's gonna be this region above our line. Yes, let's try it here. Now let's move on to our final area. OK? Where it was, if we make a note here, so you know what? We need some new colors, we'll get back to that. But let me, let's put that in green here probably. OK. And now we want to show where X minus yex minus Y is less than, or equal to three. And that was our, that was our fourth inequality. Now to figure out what this line looks like, it's not going to be a vertical or horizontal line. But we can use possible values for X and Y to help us solve. OK. So let's say when X is zero now at X equals zero, then zero minus Y will be less than, or equal to three. OK? Or rather I missed a step here. Please forgive me. But first we'll let X minus Y equal three. OK? So that we have our line, we know the points on the line and no at X equals zero. OK? Then zero minus Y will be equal to three, which means then that Y equals negative three. So zero, negative three will be on our line. OK? So that's zero, negative three which will be right here. OK? Zero, negative three. No, let's try to find the other point by letting Y be equal to zero at Y equals zero. Then we have X minus zero, equal to three which tells us then that X is going to be equal to three. Therefore 30 is also going to be a point on the line which 30 will be right here. OK? So that's 30. No, we have both those points, we can draw our line because remember our line, we can draw a line once we have two points and now we have our line here. OK? And remember our inequality was X minus Y is less than our equal to three. So the value that will be less than our equal to three will be above the line. OK. In other words, it's going to be this shaded region up here. Let me draw that properly again. OK. It's gonna be everything that's above this line. OK? And we know that that's correct. OK. Let me just finish shading here. We know that that is correct because if we take a test point, OK. So let's take a test point. Let's take uh 00. OK, which is in that shaded region. The question is, is zero minus zero, less than equal to three. Well, yes, it is because zero minus zero equals zero and zero is less than three. So since this point satisfies the inequality, that means it's in the correct shaded region. Know that we've plot our four inequalities, we need to find a solution set. How can we do that? Well, the solution set is going to be the intersection of the graphs and we can find the intersection of the graphs by simply identifying where all of the colors match with a region where all of the colors are. Now, if we look at our graph here, we can tell that all of our colors are going to be or all of our colors end up in this specific region. OK. Let me write properly here. We can tell that all of our colors will be here. OK. So that means then this area that we've highlighted in black here, this area is going to be our shaded region. This is our solution set. If we look back at our answer choices. OK, then notice that this region is the same region that we have shaded in answer choice. A therefore, a is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Graph the solution set of each system of inequalities.
$x\le4$
$x\ge0$ $y\ge0$
$x+2y\ge2$

Key Concepts

Inequalities

Graphing Linear Inequalities

Systems of Inequalities

Watch next