Graph the solution set of each system of inequalities.

Question

$3 x minus 2 y is greater than or equal to 6$

$x + y \leq - 5$

$y is less than or equal to 4$

Accepted Answer

Welcome back everyone. In this problem, we want to graph the solution set of two X minus three Y is greater than or equal to negative 22 3, X plus Y is less than, or equal to negative 13 and Y is less than or equal to seven. For our answer choices, we have our possible solutions at A B. See and D says there is no solution set and below those I have put a graph, OK. That will help us to sketch to get an idea of what our answer will look like so that we can compare. So let's go ahead and see if we can find a solution set. And for reference, let's just make note here that our equations are, are sorry. Our inequalities rather R two X minus three Y is greater than, or equal to negative 22 three X plus Y is less than, or equal to negative 13 and that Y is less than or equal to seven. OK? No, let's go ahead and solve by trying to graph each solution, each graph, each inequality rather that is, so let's start with two X minus three Y is greater than or equal to negative 22. Now notice that this inequality like the, the others are linear inequalities, which means that their border will be a line. So if we can graph the linear equations, then we should be able to shade the area that our inequality satisfies. So starting with this one, let's let two X minus Y sorry two X minus three Y OK be equal to negative 22. Now we can solve by first substituting XY zero in our equation at X equals zero. Then our equation becomes two multiplied by zero minus three, Y equals negative 22. That's going to be zero minus three, Y equals negative 22. In other words, negative three, Y equals negative 22 which means then that Y is going to be equal to negative 22 divided by negative three or positive 22 divided by three. Therefore, that tells us that zero and 22 3rd is going to be a point on our line for two X minus three, Y equals negative 22. So let's plot that because we know that 22 3rd is basically seven and a third. OK? So it's gonna be a little bit after seven for our Y coordinate. So that's zero and 22 3rd. That's gonna be probably somewhere here. OK. Let's try to plot on another point. So let's do it when Y equals zero. Now at Y equals zero, then our equation becomes two X minus three, multiplied by zero equals negative 22. So we get two X to be equal to negative 22 which means that X is going to be equal to negative 22 divided by two or negative 11. So this tells us that negative negative 11 0 is also a point on our line. So if we go to our graph, negative 11 0 should be somewhere here. OK? And now we can go ahead and connect those points to plot our graph. So no, we, we may not get it perfect. But remember the main thing we need is to compare it to the solutions that we have. OK? But when we do it, it should look something like this. OK? Let me extend it on the other side, right? So it should look something like that. And now remember that it's two X minus three Y greater than or equal to negative 22. So it's a solid line. It includes all of those points and where it is greater than or equal to that means we can shade the region which is below our line. OK? We can share the region below our line. So it's gonna be this shaded region on our graph. I'll try to shade it as best as I can. And now we know that it's this shaded region on our graph. OK? Because if we were to test, if we were to take some point, OK, that's in this region. Let's say that we take the 0.00 then if we substitute zero into our equation, are we saying that two multiplied by zero minus three, multiplied by zero is greater than, or equal to negative 22? Well, two multiplied by 003, multiplied by 00 and zero is indeed greater than equal to negative 22. So we know that we've shared the correct region. Now let's move on to our next equation. Our next inequality rather three X plus Y is less than, or equal to negative 13. Let's let OK, three X plus Y be equal to negative 13. OK? And no at X equals zero at X equals zero. Then three multiplied by zero plus Y will be equal to negative 13, 3 multiplied by 00. So that means Y equals negative 13 and thus zero, negative 13 is going to be a point on that line. So let's go to zero negative 13. It's gonna be somewhere down here. This blue point that I just put, let me write that here zero, negative 13. Next, let's let Y be zero at Y equals zero. OK? Then we know OK, that we have three X plus zero be equal to negative 13. Thus three X equals negative 13, which means that X is going to be equal to negative 13 divided by three. Therefore negative 13 3rd are negative four and a third and zero are also going to be on our line so we can go back to our graph and plot negative four and a third zero, which will be right here. OK? Now that we have these two points as well, we can go ahead and plot. OK? So let's draw our line here and we know that, let me put that a little bit better. We know that it's gonna look something like this. OK? Um Let's just extend that line down here as well. So now we have our area, OK? Or we have our line, we know that it includes, OK? It includes everything on the line and that shaded region is going to be in this direction. These are all the points that satisfy our inequality. OK. So this region that's shaded here. No, we just need to find our final region. OK? And that is oops, let me take that out here. We just need to find a final region where Y is less than or equal to seven. Now, if we think about it, this is also going to be a straight line, but we don't necessarily need to solve. OK? Because what this is basically telling us, let's push that in yellow here. What this is basically telling us is that if we graph a solid horizontal line where all of the Y coordinates are seven, which would be this line, OK? It's gonna be somewhere here, then we know the shaded region is going to be below the horizontal line. That is where we have Y values less than seven. So if we put that in yellow, it's going to be this shade all of the shaded region here. So no, this is going to be our graph for why is less than or equal to seven? Now, here is the big question. Where is our solution set? In other words, where is the intersection of our graphs? Well, if we take a look, we notice that in this region, all three colors exist in this region. So that tells us that this is where our graph intersects. OK. So this region is going to be our solution set and know if we go back to our answer choices. OK. If we compare it to our answer choices that we have that tells us then that B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Graph the solution set of each system of inequalities.
$3 x minus 2 y is greater than or equal to 6$
$x + y \leq - 5$
$y is less than or equal to 4$

Key Concepts

Graphing Linear Inequalities

System of Inequalities

Interpreting Inequality Symbols and Boundary Lines

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