Understanding Graphing Linear Inequalities

By the end of this video, you're going to be graphing linear inequalities like they're nothing. And in about 30 seconds, here's how this video is going to get you there. We're going to be graphing five linear inequalities in this video, and these problems are just going to get harder and harder as we go. And eventually, we'll talk about other things like points being in or outside of something called the solution set. So we'll talk about what that is. And then we'll also have to deal with lines that are not quite in y equals mx plus b4. So we're going to do all that. and then I'm gonna give you a problem to try and answer in the comments. And by that point, it should honestly be breezy. And if you're looking for the notes for this video, and remember when I say that, I'm talking about printable notes here. These are printable notes with a QR code attached, and they have timestamps for all the problems that we do in this video. It's literally too good to pass up on. So if you're looking for those notes, they're gonna be right in the description, and by the way, uh, they're free. In return, you can like the video. Alright, enough talking, let's get into this. So we're going to start off with y being greater than or equal to negative 2. And so how are we going to graph something like this? Well, we're going to start by graphing line y equals negative 2. And so let's do that. y controls how far up or down we go. That's what the y-axis does. And if y is negative 2, then that means we're moving down by 2. And all of the points where y is negative 2 are going to fall on this line right here. So that's the first step done. but you'll see in this inequality that y can not only equal negative 2 but can also be greater than negative 2 as well and so we don't just want the points where y is equal to negative 2 we also want the points where y is greater than negative 2 and that's all the points above this line so we just shade upwards now if you pick any of these points here these are all points where y is greater than or equal to negative 2. And these are points where y is not greater than or equal to negative 2. Moving on to our next graph, we have x is less than 4. So we're going to start this by graphing the line x is equal to 4. And this time, instead of using a solid line like we used last time, you can see that this was a solid line here, we're going to end up using a dotted line. And that is because the x here cannot actually equal 4. It can only be less than 4. You'll see that in a second. But x controls how far left. or right you're going. And so if we're going x equals 4 that's 4 to the right. So that means that we're drawing this line right here. And remember I said this line was going to be dotted so I'm going to change that to a dotted line. So now we have to figure out how we're going to shade here and since we're looking for all the values of x that are less than 4 that's going to be all the x values over here to the left of the line. So we shade that right in and that is the graph 4x is less than 4. Moving on to y is greater than or equal to x plus 3, this is a fallen line in y equals mx plus b form. So, let's graph it. We'll graph the line y is equal to x plus 3. And this is in y equals mx plus b form, as I said. And what is the slope? The slope is going to be whatever number is out in front of this x here, and you can picture that as being a 1. The reason for that, because 1 times x is just x. Just as 1 times 5 is x. is 5. It's the same idea. So that's another way that we can write x. We can write it as 1x, and there it's easy to see that your slope is 1. Your y-intercept is 3. And now we have the two pieces we need to graph this line. But is this line going to be a solid line or a dotted line? Well, that depends on this inequality here. It is a greater than or equal to sign, and the or equal to part means that it's going to be a solid line because we don't just care about where y is greater than x plus 3 we also care about where y is equal to x plus 3. so let's go and graph with a solid line so we're going to go up three that's what our y intercept tells us so we're going to start here and then we're going to go with a slope of one that means we go up one over one up one over one and now we just need a line that'll connect this so now that our line has been drawn we just need to figure out where we're going to be doing the shading so look at our inequality It's all the y's that are greater than or equal to x plus 3. Y's being equal to x plus 3 is what we just covered by drawing this line. But if we want all the y's that are greater than x plus 3 as well, then what we're going to do is shade up. We're going to shade all of these y's up here that are higher than that line x plus 3. And there we go. We have our third graph. For problem 4, we're going to be graphing. another one of these inequalities but we're also going to be talking a little bit about points that are in or not in something called the solution set so we'll talk about that after we finish graphing this inequality for graphing this inequality we can see here that the slope is 2. It's the number on x, and the y-intercept is negative 4. And now all we need to do before we start graphing this line is figure out, is this going to be a solid line or a dotted line? And well, it's all dependent on that inequality that I circled right there. Can y be equal to 2x minus 4? And no, it can't. It can only be less than 2x minus 4. than 2x minus 4. And that means we're going to use a dotted line, because we don't want to include any of those points where y is actually equal to 2x minus 4, because that's the line that we're about to graph after all. So we're going to start by going 4 down to get to our y-intercept of negative 4. And then remember that slope is rise over run. And here, the slope is 2. And we can write 2 as 2 divided by 1, right? 2 divided by 1 is 2. So that works. And what that means is that for every 2 that we go up, we're going to go over by 1. So we go up by 2, over by 1, up by 2, over by 1. And now we just need a line that will connect these points. And remember, this is a dotted line because y cannot equal to x minus 4. Any point that's on this line is where y will equal to x minus 4, and we don't want to include that. So now where are we going to be shading? Well, we're looking for all the values of y. that are less than 2x minus 4, that are lower than it. So that's why we're going to be shading down. And that's it. Our line has been graphed. Now, here's the question, though. Are these points going to be in our solution set? And actually, what even is the solution set? Well, the solution set is all of the xy pairs, all of your points that are going to satisfy this inequality. And that's going to be in our shading here. And if this was a solid line, it would also be on the line. But all right, let's ask, is 5 comma 4 in the solution set? Well, one of the ways that we can figure that out is by looking at the graph. Let's look at 5 comma 4. 1, 2, 3, 4, 5 over and 1, 2, 3, 4 up. That's this point here. And that is in our shading. So yeah, that's going to be in our. solution set. So we write yes. What about 0 comma negative 1? Well, 0 means that we're not going over left or right. And the negative 1 means we're going down 1. So that's this point right here. And that point is not in our shading. So it's not in the solution set. But what about if we had a point on the line? Like this point right here is on the line. That's the point. It looks like 3 comma 2. Well, no, that's not going to be in our solution set because this is a dotted line. If it was a solid line, we'd include it. So that's one way that you can figure it out. But let's say you didn't have the graph. Could you still figure this out? Yeah, because all of these points are supposed to work in the inequality. So you can plug these points into the inequality and see if they work. That inequality there is y is less than 2x minus 4. Well, if you plug in the point 5 comma 4 in that point, 5 is the x, 4 is the y, then what we're going to get is 4 is less than 2 times 5 minus 4. minus 4. And we'd expect this to work. So will it? Well, 4 is less than here. 2 times 5, that's 10. Minus 4 is 6. And is 4 less than 6? Yes. So it does end up working. But now let's try the other point, 0 comma negative 1. In that, 0 is the x, negative 1 is the y. So we get negative 1 is less than 2 times 0 minus 4. that one we get a negative one on the left hand side and that needs to be less than two times zero is zero minus four is negative four and is negative one less than negative four no it's not if you look on a number line here's zero here's negative one here's negative four negative four is more negative than negative one and therefore what's actually true is that negative four is less than negative one so if you ever get confused with that look on a number line it'll help you out so those are are the two ways that you can tell whether something is in the solution set or not. With that being said, let's move on to our last problem for this video. And for this one, y is not by itself here. We don't have this in y equals mx plus b form. So let's do that. Let's figure that out. Let's get y by itself on the left-hand side. To start doing that, we need to subtract x on both sides. And if we do that, we'll get 3y is less than or equal to, you could write this as 6 minus x if you want to. I'm personally going to write this as negative x plus 6. And that's just the commutative property, right? 6 minus x is equal to negative x plus 6. It's just nicer to see it this way because that's going to look more like y equals mx plus b4. So, the only other thing I want to note before we start dividing by 3 on both sides is that notice how the inequality did not flip here. I subtracted on both sides. side. The only time the inequality flips is when you multiply or divide by a negative number. That's the only time, okay? Not when you subtract. So dividing by three on both sides, if I want to divide this whole thing by three, I can just divide each of these terms by three. What I end up getting is that y is less than or equal to a negative x over 3 plus a 6 over 3, which 6 divided by 3, that's 2. And so this is my inequality. Now, what I can do to make the slope a little bit more clear. negative x over 3 that's the same thing as bringing the X down here and replacing it with a 1 it's the same thing as negative 1 3rds X if you write it like that instead then the slope becomes a lot more clear we can see here that the slope is is negative 1 3rds, and the y-intercept is 2. So let's start graphing that. And well, are we going to use a solid line or a dotted line? Well, we're going to use a solid line because y can equal negative 1 3rds x plus 2. It's less than or equal to. So let's go up 2 to get to our y-intercept. And we'll use our rise over run. We're going to go down by 1 every time we go over by 3. and that'll give us our points now we just need to connect them and the last thing that we need to do here is figure out our shading are we going to be shading up or down here well it's all the y's that are less than or equal to this line so we're looking for all the y's that are lower than this line and that's going to be everything right here and there Our last line for this video has been graphed. So that's graphing linear inequalities in a nutshell. And if you feel pretty comfortable with this at this point, then here's a problem for you to try and answer in the comments. So this problem says to graph x minus 2y is less than 8. list one point that's in the solution set and one point that's not in the solution set. So of course you can't actually graph the line in the comments. So if you want to answer it in the comments as well as listing one point that's in the solution set and one point that's not, then what you can do is tell me in the comments whether this is going to be a solid or a dotted line. Tell me what the slope of this line is going to be. Tell me what the y-intercept of this line is going to be and whether we're going to be shading up or down So try that out. Let me know what your answer is in the comments And if you have any questions on anything we talked about in this video again Let me know in the comments and i'll try to get back to you when I can Now remember that the notes for this video the printable notes are linked right in the description. They're a great resource. They're free. And so you might as well snag one just so you have. Lastly, make sure that you're subscribed to this YouTube channel. I would say more about this, but I'm kind of late for dinner right now. So I'm gonna get going. But yeah, that's gonna do it for this video, guys. And I will see you soon.

And eventually, we'll talk about other things like points being in or outside of something called the solution set. So we'll talk about what that is. And then we'll also have to deal with lines that are not quite in y equals mx plus b4.

So we're going to do all that. and then I'm gonna give you a problem to try and answer in the comments. And by that point, it should honestly be breezy. And if you're looking for the notes for this video, and remember when I say that, I'm talking about printable notes here. These are printable notes with a QR code attached, and they have timestamps for all the problems that we do in this video.

It's literally too good to pass up on. So if you're looking for those notes, they're gonna be right in the description, and by the way, uh, they're free. In return, you can like the video. Alright, enough talking, let's get into this. So we're going to start off with y being greater than or equal to negative 2. And so how are we going to graph something like this?

Well, we're going to start by graphing line y equals negative 2. And so let's do that. y controls how far up or down we go. That's what the y-axis does. And if y is negative 2, then that means we're moving down by 2. And all of the points where y is negative 2 are going to fall on this line right here.

So that's the first step done. but you'll see in this inequality that y can not only equal negative 2 but can also be greater than negative 2 as well and so we don't just want the points where y is equal to negative 2 we also want the points where y is greater than negative 2 and that's all the points above this line so we just shade upwards now if you pick any of these points here these are all points where y is greater than or equal to negative 2. And these are points where y is not greater than or equal to negative 2. Moving on to our next graph, we have x is less than 4. So we're going to start this by graphing the line x is equal to 4. And this time, instead of using a solid line like we used last time, you can see that this was a solid line here, we're going to end up using a dotted line. And that is because the x here cannot actually equal 4. It can only be less than 4. You'll see that in a second.

But x controls how far left. or right you're going. And so if we're going x equals 4 that's 4 to the right. So that means that we're drawing this line right here.

And remember I said this line was going to be dotted so I'm going to change that to a dotted line. So now we have to figure out how we're going to shade here and since we're looking for all the values of x that are less than 4 that's going to be all the x values over here to the left of the line. So we shade that right in and that is the graph 4x is less than 4. Moving on to y is greater than or equal to x plus 3, this is a fallen line in y equals mx plus b form. So, let's graph it. We'll graph the line y is equal to x plus 3. And this is in y equals mx plus b form, as I said.

And what is the slope? The slope is going to be whatever number is out in front of this x here, and you can picture that as being a 1. The reason for that, because 1 times x is just x. Just as 1 times 5 is x. is 5. It's the same idea. So that's another way that we can write x.

We can write it as 1x, and there it's easy to see that your slope is 1. Your y-intercept is 3. And now we have the two pieces we need to graph this line. But is this line going to be a solid line or a dotted line? Well, that depends on this inequality here. It is a greater than or equal to sign, and the or equal to part means that it's going to be a solid line because we don't just care about where y is greater than x plus 3 we also care about where y is equal to x plus 3. so let's go and graph with a solid line so we're going to go up three that's what our y intercept tells us so we're going to start here and then we're going to go with a slope of one that means we go up one over one up one over one and now we just need a line that'll connect this so now that our line has been drawn we just need to figure out where we're going to be doing the shading so look at our inequality It's all the y's that are greater than or equal to x plus 3. Y's being equal to x plus 3 is what we just covered by drawing this line.

But if we want all the y's that are greater than x plus 3 as well, then what we're going to do is shade up. We're going to shade all of these y's up here that are higher than that line x plus 3. And there we go. We have our third graph. For problem 4, we're going to be graphing. another one of these inequalities but we're also going to be talking a little bit about points that are in or not in something called the solution set so we'll talk about that after we finish graphing this inequality for graphing this inequality we can see here that the slope is 2. It's the number on x, and the y-intercept is negative 4. And now all we need to do before we start graphing this line is figure out, is this going to be a solid line or a dotted line?

And well, it's all dependent on that inequality that I circled right there. Can y be equal to 2x minus 4? And no, it can't.

It can only be less than 2x minus 4. than 2x minus 4. And that means we're going to use a dotted line, because we don't want to include any of those points where y is actually equal to 2x minus 4, because that's the line that we're about to graph after all. So we're going to start by going 4 down to get to our y-intercept of negative 4. And then remember that slope is rise over run. And here, the slope is 2. And we can write 2 as 2 divided by 1, right? 2 divided by 1 is 2. So that works. And what that means is that for every 2 that we go up, we're going to go over by 1. So we go up by 2, over by 1, up by 2, over by 1. And now we just need a line that will connect these points.

And remember, this is a dotted line because y cannot equal to x minus 4. Any point that's on this line is where y will equal to x minus 4, and we don't want to include that. So now where are we going to be shading? Well, we're looking for all the values of y. that are less than 2x minus 4, that are lower than it.

So that's why we're going to be shading down. And that's it. Our line has been graphed.

Now, here's the question, though. Are these points going to be in our solution set? And actually, what even is the solution set?

Well, the solution set is all of the xy pairs, all of your points that are going to satisfy this inequality. And that's going to be in our shading here. And if this was a solid line, it would also be on the line.

But all right, let's ask, is 5 comma 4 in the solution set? Well, one of the ways that we can figure that out is by looking at the graph. Let's look at 5 comma 4. 1, 2, 3, 4, 5 over and 1, 2, 3, 4 up.

That's this point here. And that is in our shading. So yeah, that's going to be in our. solution set.

So we write yes. What about 0 comma negative 1? Well, 0 means that we're not going over left or right.

And the negative 1 means we're going down 1. So that's this point right here. And that point is not in our shading. So it's not in the solution set.

But what about if we had a point on the line? Like this point right here is on the line. That's the point. It looks like 3 comma 2. Well, no, that's not going to be in our solution set because this is a dotted line.

If it was a solid line, we'd include it. So that's one way that you can figure it out. But let's say you didn't have the graph.

Could you still figure this out? Yeah, because all of these points are supposed to work in the inequality. So you can plug these points into the inequality and see if they work.

That inequality there is y is less than 2x minus 4. Well, if you plug in the point 5 comma 4 in that point, 5 is the x, 4 is the y, then what we're going to get is 4 is less than 2 times 5 minus 4. minus 4. And we'd expect this to work. So will it? Well, 4 is less than here. 2 times 5, that's 10. Minus 4 is 6. And is 4 less than 6?

Yes. So it does end up working. But now let's try the other point, 0 comma negative 1. In that, 0 is the x, negative 1 is the y.

So we get negative 1 is less than 2 times 0 minus 4. that one we get a negative one on the left hand side and that needs to be less than two times zero is zero minus four is negative four and is negative one less than negative four no it's not if you look on a number line here's zero here's negative one here's negative four negative four is more negative than negative one and therefore what's actually true is that negative four is less than negative one so if you ever get confused with that look on a number line it'll help you out so those are are the two ways that you can tell whether something is in the solution set or not. With that being said, let's move on to our last problem for this video. And for this one, y is not by itself here.

We don't have this in y equals mx plus b form. So let's do that. Let's figure that out.

Let's get y by itself on the left-hand side. To start doing that, we need to subtract x on both sides. And if we do that, we'll get 3y is less than or equal to, you could write this as 6 minus x if you want to. I'm personally going to write this as negative x plus 6. And that's just the commutative property, right?

6 minus x is equal to negative x plus 6. It's just nicer to see it this way because that's going to look more like y equals mx plus b4. So, the only other thing I want to note before we start dividing by 3 on both sides is that notice how the inequality did not flip here. I subtracted on both sides. side. The only time the inequality flips is when you multiply or divide by a negative number.

That's the only time, okay? Not when you subtract. So dividing by three on both sides, if I want to divide this whole thing by three, I can just divide each of these terms by three.

What I end up getting is that y is less than or equal to a negative x over 3 plus a 6 over 3, which 6 divided by 3, that's 2. And so this is my inequality. Now, what I can do to make the slope a little bit more clear. negative x over 3 that's the same thing as bringing the X down here and replacing it with a 1 it's the same thing as negative 1 3rds X if you write it like that instead then the slope becomes a lot more clear we can see here that the slope is is negative 1 3rds, and the y-intercept is 2. So let's start graphing that.

And well, are we going to use a solid line or a dotted line? Well, we're going to use a solid line because y can equal negative 1 3rds x plus 2. It's less than or equal to. So let's go up 2 to get to our y-intercept.

And we'll use our rise over run. We're going to go down by 1 every time we go over by 3. and that'll give us our points now we just need to connect them and the last thing that we need to do here is figure out our shading are we going to be shading up or down here well it's all the y's that are less than or equal to this line so we're looking for all the y's that are lower than this line and that's going to be everything right here and there Our last line for this video has been graphed. So that's graphing linear inequalities in a nutshell.

And if you feel pretty comfortable with this at this point, then here's a problem for you to try and answer in the comments. So this problem says to graph x minus 2y is less than 8. list one point that's in the solution set and one point that's not in the solution set. So of course you can't actually graph the line in the comments. So if you want to answer it in the comments as well as listing one point that's in the solution set and one point that's not, then what you can do is tell me in the comments whether this is going to be a solid or a dotted line.

Tell me what the slope of this line is going to be. Tell me what the y-intercept of this line is going to be and whether we're going to be shading up or down So try that out. Let me know what your answer is in the comments And if you have any questions on anything we talked about in this video again Let me know in the comments and i'll try to get back to you when I can Now remember that the notes for this video the printable notes are linked right in the description. They're a great resource.

They're free. And so you might as well snag one just so you have. Lastly, make sure that you're subscribed to this YouTube channel. I would say more about this, but I'm kind of late for dinner right now. So I'm gonna get going.

But yeah, that's gonna do it for this video, guys. And I will see you soon.

Transcript for:Understanding Graphing Linear Inequalities

Transcript for:
Understanding Graphing Linear Inequalities