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Plotting inequalities on a number line

Graph inequalities on a number line, using filled in points at the boundary for ≤ or ≥ and arrows in the appropriate direction. Make sense of the inequalities in context. Created by Sal Khan.

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Video transcript

- [Instructor] We're told that Pierre has 48 minutes until he needs to get ready for his dance lesson. Graph how many minutes he can spend playing with his pet, before getting ready. If you are so inspired, I encourage you to be so inspired, pause the video and see if you can work through that. All right, now let's think through this together. So first of all, before I even graph it, I could say, let's say that M is how many minutes he is playing with his pet, before getting ready. So let's just call it, that's what M is going to be equal to. It's how many minutes he spends. And so how would M relate to the 48 minutes? Well, he has 48 minutes, so he could spend anywhere from zero, up to and including 48 minutes. So you might say M is less than 48. But it's not just less than, it could even be exactly 48 minutes. As long as immediately after that, he gets ready for his dance lesson. So M is less than or equal to 48. So if I were to put that on a number line, let me construct a number line like this. And let me put this at zero right over here. And let's say this is 10, 20, 30, 40, 50, 60, keep going. This could be 70. This is going to be essentially the values of M that are going to be okay, as long as he's prepared for his dance lesson. M is less than or equal to 48. So what we can do is, we can go to 48, which is, let's see, this would be 45, so 40 eight's gonna be right around there. So that's 48. And since it can be equal to 48, we're gonna fill in the circle. If it just said purely less than 48, but not less than or equal, then we would put an open circle here. But because it's less than or equal to, we're going to include 48 right over there. I'll write the number 48 to make it clear what that is, and less than or equal to. So it's gonna be all of the values less than that. And so it would look something like this. I'm doing that light blue color. It would look something like then, like that. And if we wanted to be clear that we're including zero, we could actually put a dot here as well. It wouldn't make sense to go and include negative values as well. But let's do another example, a different example. So here, we are told that the Harris family needs to heat their leftover gumbo, to a minimum of 74 degrees Celsius, to be sure it is safe to eat. Graph temperatures to which they could heat their food, so that it is safe to eat. So once again, pause this video, and see if you could think about that, before we do this together. All right, well let's imagine, let's see, maybe we'll say T for temperature. So T for temperature. And let's say T is the temperature that they heat to, temperature that they heat their gumbo to, that they heat to. And now let's do a number line. We see, well, before I even do number line, let me express it as some kind of an inequality. So they need a minimum of 74 degrees Celsius. So that means it has to be at least 74, or higher. So that means T is not just greater than 74 degrees Celsius. It can also be exactly 74 degrees Celsius, 'cause it says as long as it's a minimum of 74. So if it's exactly 74, that is apparently going to be safe to eat. And anything higher than that, is also going to be safe to eat. At some point you can get to such a high temperature, that you essentially turn your food into ash. It might not be a delicious gumbo anymore, but it would probably be safe to eat, if you're just eating gumbo powder of some gumbo ash I guess. Actually, I'm not sure if that's safe to eat. But let's just assume it is. I don't recommend doing that. But let's put this on the number line. So let me, and actually here, actually, let me do it in white so that I can, in that reddish color, I can actually put the values that we care about. And we can have negative temperature, if we're talking about degrees Celsius. So let's say that this is zero degrees, this is 10, 20, 30, 40, 50, 60, 70. Let me label that one. That's 70 degrees. This would be 80 degrees right over here. If we wanted to, this would be negative 10, negative 20. And we have to be greater than, or equal to 74 degrees Celsius. If we just had T is greater than 74 degrees Celsius, we would go to 74, which would be right around there. We put an open circle, and then we would go greater than that. So that's if T was strictly greater than 74 degrees Celsius. But it's greater than or equal to. And so because of that, we are going to fill in this dot right over there. And to be clear, that is at 74 degrees and we're done.