Graphing Systems of Inequalities - Video Tutorials & Practice Problems

Linear Inequalities

Hey, everyone, welcome back. So previously, in other videos, we've seen how to graph a one dimensional inequality, something like X is greater than or equal to one. It was pretty straightforward. You just find the point X equals one on a number line and then you would just sort of figure out all of the numbers that satisfy this inequality and it would be everything to the right of that point. But we're gonna do the exact same thing here for a two dimensional inequality. It's just that now our equations that we've been dealing with have multiple variables like Y and X. So the whole idea is that we're gonna have to plot inequalities now on a two dimensional graph instead of just a one dimensional number line. But I'm gonna show you that it's pretty straightforward and I'm gonna show you step by step way to do this. So let's go ahead and take a look here. So if I had to graph the inequality Y or sorry, X is greater than or equal to one on a two dimensional number line, let's see how this would work here. Now, I've already seen with the equation X equals one looks like, remember on a one on a two dimensional graph, this will actually be a vertical line. It's kind of weird because on a one dimensional number line, it's just a point. But remember now we have like sort of a we have a Y axis. So you almost could kind of like extend this up and down and you'll see that you'll get a vertical line. So the keyhole I idea here is that to graph an inequality first, you actually have to graph what the corresponding line is. But now we have to look at sort of how to shade the parts of the graph that will satisfy this inequality. And so what you're gonna do here is you're gonna shade the side of the graph with points that make the inequality true. You remember you're just gonna sort of uh figure out the points that make this inequality a true statement. And what I can do here is just sort of like pick random points like uh let's say, for example, I picked this 0.2 comma zero. If I take these X and Y coordinates and I plug them into this inequality, we'll see that we get true statements for this. For example, the X coordinate for this is two. So two is greater than or equal to one that is a true statement. But now we also have like an infinite number of Y values to consider. We can pick like a point over here, this is four comma three. When you plug it into this inequality, you'll also get a true statement. So the key difference here is that we're not just gonna shade the axis like one line, we're actually gonna have to shade everywhere in this graph with points that satisfy this inequality. And for this particular line, it's actually gonna be all of these points that are to the right of this inequality. So it kind of makes sense for one number line, you just had a line or a two dimensional graph, we actually have like a like an area, a region of points that satisfy this inequality, right. So that's all there is to it. All the points on this side of the graph are where X is greater than one. So that satisfies this inequality, all the points over here are less than one. So if you had something like Y is less than X, for example, we could graph this pretty simply, all you'd have to do is just graph, the corresponding line Y equals X. Now, one thing I wanted to point out here is the symbols are different. Notice we have a less than symbol whereas we had a greater than, or equal to symbol. So if you have equations with the uh less than, or equal to or greater than equal to symbols, you draw a solid line. The way I like to think about this is if you see a solid bar underneath the symbol, then you draw a solid line. But if you have something like less than, or greater than, or less than, then you draw a dash line. So Y is less than X is gonna look like Y equals X. But we're gonna have to draw it with a dash line. So it's basically gonna look something like this. All right. Now, these are, these examples are pretty straightforward, but sometimes you're gonna get more complicated uh equations like something like Y is greater than two X minus four. So I actually want to sort sort of show you a step by step process of how to graph those inequalities. Let's get started here. So the first thing you wanna do is you're gonna graph the solid or dash line depending on the symbol by switching the inequality symbol with an equal sign. So remember that this graph of this symbol over here means that we're gonna be drawing a dashed line. And the way we graph this is by graphing the corresponding line, what is Y equals to X minus four look like? Well, it just looks like a line that co that goes through the Y intercept of negative four and has a slope of two. So it's gonna look something like this, we have to use a, so a dash line for this. So this is gonna be what that graph looks like. It kind of just like sketched out. Um So that's the first step, right? So the second thing we have to do is we have to figure out which points will satisfy in inequality with X is greater than or equal to one. It was pretty simple because we just highlighted everything to the right of this graph. But for this, it's gonna be a little bit tricky, right? So is it gonna be this side over here? Is it gonna be this side? How does it actually work out for different angles and different steepness is? So the second step is you're actually, you basically test a point on either side of the line. And the way that you test it is you just basically plug or by plugging the XY coordinates into the inequality, which is exactly like we did with this two comma zero over here. So the idea is that we're just gonna pick a point randomly. You can just pick any point that you want, it won't matter. But it's always just best to use a point on the X or Y axis because it makes the math a little bit easier. So for example, if I pick this point at random, this is gonna be zero comma one, I'm gonna test it by plugging the X and Y values into this inequality. So zero comma one, if I plug it into this inequality over here, what this says is that is one greater than two times zero minus four, right? Because that's in this inequality says over here. So let's work this out is one, in fact greater than or greater than this inequality. Well, two times zero is zero. And again, that's why using a point on the axis is easier because one of the variables sort of cancel out. So one we'll see is greater than negative four. Is that a true statement? It is one is greater than negative four. And so this is gonna be a true statement. So that's the second step sort of testing out a point. Now, the third one says that if the point makes the inequality true, like what exactly what we had in this case over here, then you're gonna shade the side that includes that point. So shade the side with that point. So on this graph here, the line two X minus four is like a barrier and we're gonna shade everything that's to that, that includes this point over here that our statement true. So really what happens is the region that makes this inequality true is actually gonna be all of this area over here. If you were to take a point at random anywhere inside this region, the X and Y values will make this inequality a true statement. Versus if you pick something in this region over here, you'll get something that doesn't make sense like five is greater than seven, right? That doesn't make any sense. All right. So the uh if you, if, if your point makes the inequality false, which is not what happened here, then you're gonna shade the side without that point. And that's really all there is to it. All right. Now, there's actually a cool shortcut here that I'll teach you, which is that if you can ever isolate y to one side of your inequality and get it in slope intercept form, which is basically what we have here. And if you have something like a greater than symbol or greater than, or equal to, then you can just shade everything that's above the line. So that's exactly what we had here. Y is greater than two X minus four. This is the line. You would shade everything that is above this line. Whereas if you had something like a, less than symbol that you would shade everything below that. All right. So that's how to graph linear inequalities. Thanks for watching and I'll see you in the next one.

Example 1

Welcome back everyone. Let's continue our journey through systems of inequalities by taking a look at this example problem here here. We're told we're given these three equations. Uh A B and C and we're told to match each inequality with its graph. So we're not actually gonna graph anything. We're just gonna sort of be matching up equations to graphs. Let's go ahead and get started here. So I've got equation A which is Y is less than or sorry, Y is greater than negative three X plus four. And then for equation B, we've got X plus Y is less than one equation C is negative three, X minus Y is greater than equal to negative four. Let's take a look at the first equation over here Y is greater than negative three X plus four. Now, in order to visualize the graph, remember the first thing you do is kind of just to sort of pretend here that there is like an imaginary equal sign. So what is the graph of negative three X plus four look like? Well, it looks like an equation of a line with A Y intercept of four. And if you look through these two graphs, there's actually two equations with Y intercepts of four graphs. Two and three. And they almost look the same except now the shading is in different areas and one solid and one is a dash line. So which one does this equation correspond to? The next thing you should look at is the sign or the symbol that we see. Because here we see a greater than symbol whenever you see a greater than a symbol or a less than symbol without the solid bar underneath. That means that we're dealing with a dashed line, not a solid line. So that means that without a doubt equation A corresponds to graph number three. So this corresponds to graph A, all right. And if you can see here, uh So we've got the dash line with a negative slope of three. And we also can see that the shaded area is everything that's above uh that line, which is exactly what the shortcut for lines tells us. Notice how this symbol here is a greater sign. So we're gonna shade everything that's above that line. All right. So now let's take a look at equation BX plus Y is less than one. So this may be hard to visualize the graph it looks like. So let's rewrite it in slope intercept form. I could just move the X to the other side by subtracting X. And this is gonna be Y is less than negative X. Plus one. So same thing over here, we've got a Y intercept of positive one and notice how, what I said in the beginning or when we were dealing with. The first equation is that these two lines have Y intercepts of four, the only one that has a Y intercept of one is gonna be this line. So this is definitely corresponds to equation B, all right. And just to sort of double check here, uh we can see that this equation tells us that we should be dealing with a solid, sorry, a dashed line because it's just a less than symbol. And that's exactly what we have. We have a dashed line, it's got a slope of negative one. And also what we can see is that because of the less than symbol, uh our shortcut tells us that we have to shade everything that's below that graph. So this is without a doubt the equation or uh the, the graph of B. And that just means that by default, this second graph over here is gonna be the equation for C. But let's understand why that works. So here, we've got this other sort of more complicated and nasty expression. One of the things that you can do here is you can sort of tell that all of these graphs here are these numbers here have a negative sign. And so one thing you can do is you can actually flip all of those two positives. But then you have to flip the inequality symbol. So one way this sort of like simplifies this actually just turns into three X plus Y is less than or equal to positive four. When you flip all the signs, you actually flip the symbol as well. And finally, what we can do is move the three X to the other side which you'll get is Y is less than or equal to negative three X plus four. That would be in slope intercept form. So again, notice how we still have that positive four of A Y intercept. We still have the same slope of negative three. But now what happens is that we have a less than or equal to. So this should be a solid line and we should shade everything that's underneath that line instead of above like we did for graph number three. All right. So that means that this is definitely the equation or the uh equation for this graph over here. All right. So that's it. Folks. Let me know. Uh let me know if you have any questions and thanks for watching, I'll see in the next video.

Graph the inequality $2x+3y$ < $6$.

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• D.

Nonlinear Inequalities

Everyone. Welcome back. So in previous videos, we saw how to graph linear inequalities, something like Y is greater than two X minus one. But now you might start to see problems in which you have nonlinear equations, things like quadratics or exponentials radicals and rational functions, things like that. I might think that it's an entirely new process, but I'm actually gonna show you here a graphing nonlinear inequalities is exactly like graphing linear inequalities. We're actually gonna follow the exact same list of steps here. So let's just jump right into this problem and see how it works. We've got the uh Y is greater than or equal to X squared minus one. So this is clearly going to be a quadratic equation, right? So we have something squared. So let's just go ahead and stick to the steps here. The first thing we do is figure out we're dealing with a solid or dash line. Let's look at the symbol we see a greater than or equal to symbol. If I see a solid line underneath, it's gonna be a solid line. Um That's gonna be my equation, right? So what is X squared minus one? Look like, well, first you're gonna have to sort of pretend that there's like an equal sign there. So Y equals X squared minus one is going to be a parabola in which the vertex is gonna be at zero, comma negative one. And then it opens up like this. So what's gonna happen is you're gonna look something like this. I'm just gonna sketch it out. It doesn't actually have to be perfect unless your professor uh really wants this to be perfect. But this is gonna be what our parable looks like, right? So that's why equals X squared minus one. Let's take a look at the second step. The second step says we're gonna test the points on either side by plugging, but actually, it doesn't need to be a line. It could be a curve or something like that by plugging X comma Y the values into the inequality. All right. So we can use uh something on the X or Y axis so we could just pick a point at random. What I'm gonna do is I'm just gonna pick this point over here. Zero comma two. That's pretty much like well within uh sort of the upper portion of the Parabola. So let's go ahead and test this out zero comma two. Remember this is just my X and Y values. So what this means here is that Y is greater than, or equal to X squared minus one. So I'm just gonna replace them. Now two is greater than or equal to zero squared minus one. So does this make a true statement? But what you'll see here is that you get two is greater than or equal to zero squared is just zero and is two greater than or equal to negative one. This actually is a true statement. Two is greater than or equal to negative one. So because this is a true statement, now we can move on to step number three, we're just gonna shade the side that includes that point. If we, our statement is true, we include that point. So really all this is here is remember the the curve or this equation here is kind of like a barrier, we can't shade anything that's on the opposite side of that barrier. So you might think, oh well, I just shade everything that's above this line over here. But that's not how it works. Because if you were to pick a point, let's say that's out here on above the equation or above the graph, but still on the opposite side of this line, you'll find that this actually won't be true. So what you have to do is you're sort of like bounded by the shape of the graph itself. So really the points that make this this inequality true are actually gonna be everything that's just above the parabola, but still kind of like on the inside of it. So it's kind of like on the inside of the bowl. All right. And that's really all there is to it. So that would be how to graph this inequality and shade uh the appropriate areas. All right. So that's how to graph nonlinear inequalities. Let's go ahead and take a look at some practice.

Graph the inequality $y$ < $2^{x}$.

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Example 2

Welcome back everyone. So just in case you were struggling with this problem, I'm here to help. So we're gonna graph this inequality X squared plus Y minus one squared is less than equal to nine. So let's get started here with our steps. First, we're gonna have to figure out whether you're dealing with a solid or dash line. And we do this by the, looking at the equation or the, the inequality symbol. So remember if there's a, if there's like a bar underneath the symbol, that means it's gonna be a solid line, that's exactly what we have here. So we're gonna have a solid line not dashed. And we're gonna have to graph this by switching the inequality symbol with an equal sign. So we're basically just gonna have to graph the inequality X squared plus Y minus one squared is equal to nine. All right. So what does this look like? You may have been trying to re sort of rearrange this and trying to solve for a while, you'll get some weird square roots. But actually, if you remember here that we've dealt with these kinds of formulas before this basically is kind of like the equation X minus H squared plus Y minus K squared is equal to R squared. If you remember this, this is the equation for circles. So circles over here have like an X squared and A Y squared equal to something on the right side. And these numbers H and K just represent the vertex. So with this looks like this equation here is, it's a circle and the center is gonna be at the coordinates that are sort of inside of the parentheses. So here, if we have not, no numbers inside the parentheses, that means it's a zero. So center is at the coordinate zero comma and then this number over here gives our K value. And remember when we see a minus sign that means the K is actually positive. So this is a center at the sorry, this is a circle with a center at zero comma one. All right. So that's what this formula is. The circle is gonna look like. So the circle center is gonna be here at zero comma one. But remember we actually need the radius of the circle and that's given to us by this information over here, the number that's on the right side. So clearly we can see that this R squared is equal to nine. And so the radius R is equal to the square roots of nine, which is just equal to three. So remember we're gonna have to do is we're gonna have to go sort of up, down and to the right and left by three units. And then we're gonna have to connect all of those points over here. All right. So this is gonna be uh 123 and we're gonna have to graph, oops, I'm gonna have to graph a circle that connects all of these points together. All right. And once they do this, it's gonna look something like this. All right. So it's sort of like my crude circle there. So remember we're gonna use a solid line for this, but this is basically what our graph is going to look like. Now, let's take a look at the second uh step over here, which is we're gonna have to test a point because even if you got this far, you may not have noticed where to shade this uh this graph. Do we always go below the circle? Do we go above it? What's going on here? So let's go ahead and test the points. Uh That's on the X or Y axis. All right. So what I'm gonna do is I'm gonna test the points. Uh I'm just gonna test this point over here, which is the 0.2 comma zero. All right. So let's test this out. So what about two comma zero? So what this is saying here is that you're gonna have to do two squared plus and then you're gonna have to do zero minus one squared is less than or equal to the number nine? All right. So let's actually just move this over here and test this out. So this is gonna be four plus and we've got zero minus one, which is negative one squared which is positive one or plus one is less than, or equal to nine. So in other words, is five less than, or equal to nine. And is it actually a true statement? So five is less than or equal to nine. So that means we're gonna have to shade the point that includes that point uh within the graph. So, but does that mean that we actually have to graph everything that's below the circle? Well, actually, no, and the way you can sort of test this out is you can sort of test out and at this point over here that is still below the circle but outside of it. And what you actually see here is that if you test this point over here, you'll see that the inequality actually fails. And so what happens here is we actually have to shade the area that's actually within the circle. So these types of inequalities are a little bit sort of trickier because you can't just shade everything that's below the circle or the center of the circle, you actually have to test all these points, which you'll see is that only the points within the circle will actually make the inequality true. All right. So that's how to graph these types of inequalities with equations of circles. Let me know if you have any questions and thanks for watching.

Systems of Inequalities

Everyone. Welcome back. So up until now, we've seen how to graph an individual inequality. Something that looks like this. The basic idea was that we would draw this inequality and test the points and then we would shade the area, an area that makes that inequality true with those points. But now we're actually gonna see some problems that have multiple inequalities. And in this case, what we're gonna do is we're just gonna repeat this process over and over again, go same list of steps that we have for as many inequalities as we have. And we're gonna see something interesting that happens. So for example, if I were to graph this inequality and let's say, just look something like this, we're gonna see is that when we graph this inequality or our, our shading area is gonna overlap with the shading area of the first graph. And what we're gonna see here is that there's gonna be this one region where all of the shadings will overlap and this will be the solution to the system of inequalities. So the whole idea here is that a graph, a system of inequalities, you're gonna plot all the lines all the curves and shade the region's making points or uh containing points that make all of the inequalities true. And to do this, we're gonna use different colors and different styles of shading. You can do this however you want. Um basically shading each curve first and then later on, we'll find the overlap. So let me go ahead and just break this down for you. We'll follow this list of steps here. Let's take a look at our first example. We've got Y is less than, or equal to negative X plus four. Let's go through our steps. We're gonna graph the solid or dashed curve. This is gonna be a solid line because we have a solid symbol underneath the equation and we're just gonna replace it with an equal sign, right? So what is Y equals negative X plus four look like it looks like it goes through the points of the Y intercept of zero comma four and it has a slope of negative one. So this is gonna look something like this. It's gonna be a solid line. So it looks like that. All right. So that's our equation or, or that's our graph. And then to test the point, we could just pick any point that's uh underneath this graph or really anywhere. So I'm just gonna pick this point over here. This is gonna be a 0.0 comma two. So if I test out this 0.0 comma two, what happens Well, this is saying that two is less than or equal to negative zero plus four. So is this actually a true statement? The zero will go away. So is two less than or equal to four? That is actually a true statement. So we also just could have used the shortcut that because of the sign here where we're just gonna graph everything that's underneath this line and we, that would have been perfectly fine. So this all of this area here, this whole thing satisfies uh the red inequality, the red equation that we have here. All right. Now let's do the same exact thing for the blue equation, right? So we did all these things, the blue equation, we're gonna graph this equation Y is greater than two X plus one. So in this case, we're gonna have is a Y intercept of one and it's gonna have a slope of two, but this one's gonna be a dash line because of the symbol. So in other words, it's gonna look something like this. So this equation is gonna look something like that. That's gonna be our line. And now let's go ahead and graph the uh inequality. Now, what we could do or sorry, shade the area. Now don't have to test the points. You actually don't have to pick a different point. You could pick the exact same point that we did over here. So what about zero comma two? Does this satisfy this inequality? What this is saying is that two is greater than uh zero plus one. Is that a true statement? Well, two in fact is greater than one. So this is a true statement and therefore we've tested the points and we're gonna shade that inequality. So in other words, for the blue equation, it's actually gonna be everything that includes this point as well. So what would we see here? We're gonna see that when we shade this equation, we're gonna see some yellow, some blue and some green. You also could have seen this use different sort of styles of hatching of, of, of, you know, crossing out. So I've seen some people do something like this and then they'll use markings like this for the blue one and you'll just find the ones, the ones where they overlap. So there's a bunch of different ways to do it. But in this case here, the point is is that in some cases or, or sorry for this region, these are gonna be the points that satisfy only just the uh inequality. The first inequality, negative X plus four, the blue area contains points. It's only gonna satisfy this inequality. But the region that contains points that satisfies all of them is gonna be the place where the shadings overlap. So that's the last step, you just shade the overlapping of all of the shaded regions. So this is gonna be the answer to your solution here. So this is your solution. All right. So that's really all there is to it, you just go through the steps multiple times to refer as many inequalities as you have. So it's pretty straightforward. Now, one thing I want to mention here is that one or more equations may actually be nonlinear. So you could have instead of two lines, you could have one in, you know, that's a problem or a circle or something, but it actually just works the exact same way. We're just gonna sort of fly through the second example over here. See that's pretty straightforward. So I'm actually gonna use some shortcuts here. So for example, if I wanted to graph this inequality Y is greater than or equal to X squared minus four, we've seen how to graph parables before. Um It's really just gonna look something like this. So X squared minus four is a Parabola that opens up at, with a vertex at zero comma negative four. And it's gonna look something like this. It looks something like that. So I'm just gonna sketch this out. It's gonna be a solid line because of this bar over here. But we've already seen that for Parabolas, if you have a greater than or equal to sign, it's gonna be everything that's above the Parabola or inside of the bowl. So you could test the point and you would find that any points inside of the shaded area over here would satisfy this inequality. So now for something like negative X plus three, then you would just graph this line and this would just be a point or a line that goes through zero comma three and has a slope of negative one. And this case would also be a dash line. So it would look so, oh sorry, a solid line. So it would look something like this. That's what the blue equation would look like. Now, for the blue equation because of the sign here, we're gonna shade everything that's underneath that line. That's kind of like our shortcut. And so it would be everything that's in this part of the graph. And what we're gonna see here is that the yellow and the blue eventually will overlap and they'll sort of be everything that's in this area over here. These points contain the solutions that satisfy both the inequalities at the same time. So this area over here. Oops. So this area over here, this is gonna be your solution. So this is the area of your solution. All right. So that's the solution to this inequality over here. Now, with just one last point here, I want to point out that some system of inequalities may actually have no solutions. We'll see what that looks like later on. So that's it. For this one, folks, let me know if you have any questions

Graph the system of inequalities and indicate the region (if any) of solutions satisfying all equations.

$3x-2y>6$

$3x-2y$ < $-4$

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Graph the system of inequalities and indicate the region (if any) of solutions satisfying all equations.

$x+y\le4$

$y\ge1$

$x>0$

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• D.