Graph the solution set of each system of inequalities.

Question

$2y+x\ge-5$

$y\le3+x$

$x\le0$

$y\le0$

Accepted Answer

Welcome back everyone. In this problem, we're given a system of inequalities two X plus Y greater than, or equal to negative five Y greater than, or equal to two X minus eight X greater than our equal to zero and Y greater than less than, or equal to zero. Sorry. And we want to graph the solution set for this system for our answer. Choices, they give us possible answers or possible solution sets. We are here, answer choice A shows a possible solution. So does answer choice B answer try C and then answer choice D tells us that the solution set is empty. Now below our answer choices I have also put a graph here that we can use to help us plot our inequalities. Because if we can plot all of those inequalities on a graph, then the solution set is going to be the area or the region rather where all of those inequalities intersect. So let's go ahead and try to do that. Let's start by trying to graph two X plus Y greater than or equal to negative five. Now to graph the inequality first, we have to draw the line for the equation that corresponds to the inequality. And then we need to decide which region which side of the line we're going to shade that satisfies the inequality. So let's let two X plus Y be equal to negative five. No, this is a linear equation. So to graph it, we only need two points and we can get those two points by substituting xry values into our equation to find a corresponding xry value to, to, to make up here. OK. So let's say that X equals zero. Then in that case, our equation would be 202, multiplied by zero plus Y equals negative five, which tells us that Y would be equal to negative five. Therefore zero, negative five is going to be a point on that line. And if we go to our graph, we can plot zero negative five. So it would fall right here. OK? Next, let's try to figure out the X value when Y is zero at Y equals zero. Our equation becomes two X plus zero equals negative five. Thus two X equals negative five. And that means X equals negative five divided by two. Therefore negative five divided by two and zero are also a point on the line. If we go back to our grid, negative five, divided by two would be negative 2.5. So that means negative 2.5 0 would be right here. And thus no, we can draw our line and we know this line is going to be a sod line because if we look at our equation, it's greater than or equal to. So it includes all of the points on the line. Now, which side of our line should we shade? Well, let's pick a test point. OK, let's test it with 00. Is 00 or sorry. Does 00? Does the coordinate 00 satisfy the inequality that is does 20 plus zero give us a result that's greater than, or equal to negative five. Well, two times zero plus zero equals zero and zero is greater than negative five. Therefore, we know that the inequality is true. So wherever the 0.00 is, that's the region we shared and we know that it's that point right here. OK? So let's just uh put our line here. OK? Let me just make this a bit brighter. OK? So we know our line is gonna go along here and we're gonna be shading this side. OK? So we have our first inequality. Now let's move on to the second inequality. OK? Now remember for a second one, it was Y greater than or equal. Let me put that in blue, Y greater than, or equal to two X minus eight. No, if we do the same thing here like we did before that is OK. Let's uh let, why equal to X minus eight? OK? Then when at Y equals zero, OK? Then zero equals two X minus it. Therefore, two X equals eight. OK? And if two X equals eight, that means X equals four. Therefore, we know that 40 is going to be a point on our line. So again, let's try to plot that on our line. OK. 40 on our graph here is 40. OK. Now what about if X equals zero? Well, when X equals zero, our equation is Y equals two multiplied by zero minus eight, which tells us then that Y is going to be equal to negative eight. OK. Therefore, zero, negative, it is also a point on our line. So let's go to zero, negative eight, which is going to be right here. OK? Now that we have those two points, we can draw our line that goes through them. OK? And now that we have our line, oops, let me try to draw that properly here. Now that we have our line, we need to figure out which side of our line we're going to shade again. Let's consider a test point. We could use the same test point, we could use 00. Now, if we test it with 00, the question is, is zero, greater than, or equal to two multiplied by zero minus eight. Well, two multiplied by zero minus eight is negative eight and zero is greater than negative eight. So that means the region where 00 is, is going to be is going to satisfy our inequality. So if we put that in blue here, OK, then this region OK. This region let me just jot along our line and then let's just shade in the rest. This region is also going to satisfy our inequality. So now we've put two of our inequalities on our graph. Now we need to figure out the other two. Now remember for the third one, the third inequality, we said that X is greater than, or equal to zero, that's going to be a straight line. Well, sorry, first of all X equals zero is a vertical line that passes through the origin. OK. So if we were to put that in red, X equals zero could be this line. And no, because it's greater than zero, that means we're going to shade in red, everything that is greater than zero along this line. OK? So let's just draw our shade here along the line. Let me just fix that. OK? And now we know that we're going to shade everything after that line. OK? So we're going to shade everything to the right of the line. And now finally, or, or, or last inequality said OK, that Y is less than or equal to zero. OK? So the line Y equals zero is the horizontal line that runs on the X axis. OK? So if we're looking for those, let me put that here. So this is Y equals zero. Likewise the one before is X equals zero. So now we're looking for everything that's below this line, everything where Y is less than or equal to zero. Of course, it includes zero. OK? So now when we do that, it's gonna be everything below this line. OK? Everything below our line here. So now that we've put all our inequalities on the graph, the question is which region satisfies all of the inequalities? Well, I like to keep it shaded because one way I like to think about it is that it's the region that has all of the colors. Now, if we look carefully at our graph, we can see that it's going to be this region right here. So it's gonna start at the original. It's gonna come down here. OK? It's gonna be formed here and then it's going to go along the Y axis. So it's going to be this area that I've shared in black. This is the region that represents the solution set. Now, if we compare this to our answer choices, we can tell then OK, we can tell that C is the correct answer. Therefore, C is the graph that shows us the solution set. Thanks a lot for watching everyone. I hope this video helped.

Graph the solution set of each system of inequalities.
$2y+x\ge-5$
$y\le3+x$ $x\le0$
$y\le0$

Key Concepts

Inequalities

Graphing Linear Equations

Feasible Region

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