Graph the solution set of each system of inequalities.

Question

$2 x plus 3 y is less than or equal to 12$

$2 x + 3 y > - 6$

$3 x plus y is less than 4$

$x is greater than or equal to 0$

$y is greater than or equal to 0$

Accepted Answer

Welcome back everyone. In this problem, we want to graph the solution set of two X plus Y is less than, or equal to +52 X plus Y is greater than negative 220 X plus six, Y is less than 30 X is greater than, or equal to zero. And Y is greater than, or equal to zero. For our answer choices. We have possible solutions for our graphs. Here's a possible solution. A possible solution B A possible solution for a sea and D says that there is no solution set. And below these, I've added our graph here a blank graph to help us plot our inequalities. Now, if we're going to figure out the solution set, we need to plot each of our inequalities on a graph and identify where, where they intersect. So let's go ahead and try to do that. Let's start with our first inequality. Let's put that in red here and recall that our first inequality was two X plus Y is less than or equal to five. OK. Now, if we're going to graph this inequality first, we need to find the line two X plus Y equals five. And then we can find inequality by figuring out where on our line we need to share it. OK? So let's let two X plus Y, OK be equal to five. And I think on our graph here, let me just identify our Y axis and our X axis respectively. No notice here that this equation is a linear equation. So to figure out what the line will look like to draw the line, we only need two points on our line. So let's substitute possible values for X to get values for Y which we can use to plot our coordinates. Let's say that X equals zero. Now when X equals zero, our equation becomes two multiplied by zero plus Y sorry plus Y equals +52, multiplied by zero is zero. So that means then that Y is going to be equal to five. Thus +05 is going to be a point on our line. We can go ahead and plot 05. Next, we can try to figure it out when Y equals zero because no at Y equals zero, then we have two X plus zero being equal to five, which means that two X equals five. And thus that tells us then that X, if we divide both sides by two is going to be equal to five divided by two. So now this tells us then that five divided by two and zero is also a point on online. So that means where X is five divided by 25 halves or 2.5 and Y is zero, that's also gonna be on our line. So that point is going to be here now that we have both points, we can plot our graph and now we can use this red line to represent our graph. OK. And let me also plot it in the opposite direction. And no, this is going to be our line that shows two X plus Y is less than or equal to five. Now this is the line two X plus Y equals five. Let's shade our proper area to show the inequality. Now, if we think about it, if we choose a test point, let's choose a test 0.00. OK. If we choose that test point, does it satisfy the inequality? In other words, there two multiplied by zero plus zero or is two multiplied by zero plus zero, less than, or equal to five. Well, two multiplied by zero plus zero is zero and zero is less than five. Therefore, it satisfies the inequality. So wherever that point is is going to be our shaded region. Thus, this tells us that our shaded region is below the line. OK? Which is going to be in this direction. OK. So first let me try to just make sure that I don't go over our line here. OK. So we have our shaded area. I mean it's not gonna be perfect, but you can see which area should be shaded. And now we can go ahead and shade the region for two plus Y is less than or equal to five. So this is our shaded region. Now let's move on to our next inequality where we know that that inequality is two X plus Y put that in blue, OK? Two X plus Y is greater than negative two, not just like with our land before we need some values for X that we can substitute to get two points and go ahead and sell. Now let's use X being, let's let two X plus Y sorry, be equal to negative two. And now let's try to figure out what our point is when X equals zero, not X equals zero, then our equation is going to be two multiplied by zero. OK plus Y greater than or sorry, equal to negative two, two multiplied by zero equals zero. So this tells us that Y equals negative two. Thus zero, negative two is going to be a point on that line. And if we go to our graph here, then zero, negative two will be this point. OK? Now what about when Y equals zero? We could use that as well when Y equals zero, then our equation is going to be two, X plus zero equals negative two. That means two, X equals negative two. And thus if two X equals negative two, then X OK must be equal to negative one. Because two multiplied by negative one equals negative two. So this also tells us that negative +10 is a point on our line. So when we plot negative 10, OK, now we have two points. OK? And since we have two points, we can go ahead and plot our line. So let's do it in this direction. OK? And then let's go in the other direction. And now we know that our line is going to be dotted. OK? So let's just put those in there. So we know our line is going to be dotted because it says it's greater than negative two. So it doesn't include the points on our line. Let me try to make this look a bit better. It doesn't look exactly how I want it. OK? I know it's gonna be a dotted line. Now, the question is where will or shaded region f again, we can use the same idea. So let's test it with 00. OK. Now, when we test it, OK, we're asking ourselves, the question is two multiplied by zero plus zero, greater than negative two. Well, two multiplied by zero plus zero is zero and zero is greater than negative two. So 00 should be in the shaded region, which means that we're going to shade the region above our line. So let's get our blue marker here and let's go ahead and share the region above the line. Ok? I'm just gonna be in this region up here. So now we're almost there, right? We just have three more inequalities to go next. Let's move on to our next. The, the other inequality 20 X plus six Y is less than 30. OK? Now, for this inequality, OK. 20 eggs. Let's put that in black here. 20 X plus six Y is less than 30. OK? Now we know that for this graph, OK? Uh for time's sake, we won't show all the working but when X is zero, OK, the Y value is going to be five. So +05 is going to be on the line of 20 X plus six Y equals 30. And when Y is zero, OK, X is equal to three halves. So then that tells us that three halves, zero is also going to be on the line that it forms. So again, if we plot those points, that's 05, OK, which is gonna be up here and three halves and zero, that's 1.5 0, which is gonna be here. And now if we go ahead, connect our line, OK, it's gonna come down here in this direction. And now we know that because again, because it is less than, then that's also going to be a dotted line. So we can put our a dotted line here. And now if we were to test our points, OK? If we were to test our points, we would know that the area below our graph is going to be our shaded region that is where it's going to satisfy our inequality. So we can put that in yellow, OK? And we can share the region below our lines. And so it's gonna be, here's our line, right? It's gonna be the shaded region that's below our line down here. Now, we have two more inequalities left. OK? Because we know that for the rest of our inequalities, we have X is greater than I equal to zero and Y is greater than I equal to zero. Now, for both of these, we know that this is, this just tells us that our X and Y values are positive for X is greater than zero. OK? It's going to be if I use here, let me use green for both of these. This is going to be the region where X is greater than zero. OK. Right, X is greater than zero. And now where Y is greater than zero, it's going to be the region above the X axis. OK. So in other words, there's going to be this region up here. So now that we have this region, OK, if we are going to find a solution set, we have to ask ourselves, where do our grass intersect? One easy way to figure that out is to find the area where all the colors are now in which area are all the colors. Well, if we think about it here, notice that all of our colors are inside this region. OK. So there, inside this region, let me use a straight line and I'm going to highlight with a black line. OK? So they're between here, here and here, OK, where we have our dotted lines. So this area that I've shaded in black is going to be the solution set to our inequality. When we look back at our answer choices, when we compare our results with our answer choices. OK. That tells us then that C would be the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Graph the solution set of each system of inequalities.
$2 x plus 3 y is less than or equal to 12$
$2 x + 3 y > - 6$
$3 x plus y is less than 4$
$x is greater than or equal to 0$
$y is greater than or equal to 0$

Key Concepts

Graphing Linear Inequalities

Systems of Inequalities

Coordinate Plane and Quadrants

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