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Lesson 7: One-step inequalities

# Plotting inequalities

To plot an inequality, such as x>3, on a number line, first draw a circle over the number (e.g., 3). Then if the sign includes equal to (≥ or ≤), fill in the circle. If the sign does not include equal to (> or <), leave the circle unfilled in. Finally, draw a line going from the circle in the direction of the numbers that make the inequality true. Created by Sal Khan.

## Want to join the conversation?

• At can we also use <= instead of ≤ ? Will that be correct?
• If you are writing the symbol by hand, then you should use "≤". When typing on a keyboard, it is common to use "<=" as it requires no special characters other than what is already on the keyboard.
In KhanAcademy exercises, they provide a menu of special symbols so that you can use "≤".
• Hi,
Small question, I just want to know how can I get the most out of these lectures on Khan academy. Any tip or tricks welcome here ;).
Thanks
• Pause the video and have a go at the question. This will make you think and help better understand the problem. Also, you can identify any mistakes you might have made and correct them.

Do all the exercises, try to get 100%. It's okay even if you don't, the goal here is to practice and learn.

Take notes, jot down anything you feel is helpful. (I like to summarize each topic to one page so I can quickly go through everything.)

Don't give up if you don't understand something. Take a short break, maybe listen to some music. Come back and try again.
If you are stuck, you can also ask for help in the comments section.

Good luck!
• When graphing less than or less than and equal to 1500 calories/day, you shade the number line in so that it goes below zero into the negative. How can you consume a negative number of calories in a day?
• Of course, to consume negative calories you would not be gaining any calories, but losing them. Meaning you would be working out and burning them.
To answer your first question, he shaded the arrow at zero which shows it can be any negative number beyond zero as well. So we displayed that on the number line.
• I have a challenge for all Khan academy learners;

Mr. Monroe keeps a bag of small prizes to distribute to his students. He likes to keep at least twice as many prizes in the bag as he has students. The bag currently has 79 prices in it. Mr. Monroe has 117 students. How many more prizes does he need to buy.

Write the answer as a comment in an Inequality equation

For example 5000<=83x+850
• Prizes = p

Needs: 117*2p prizes.
Has: 79 prizes.
Will buy: 234p - 79p = 155 prizes.
He likes to keep atleast double the amount of prizes as students. Thus, our equation is:
`p≥155`

(Did I do it right? 🤔)
(1 vote)
• I'm confused where you got 150 from.
• What does the sign ≤ mean? I've always wondered that... I don't think it says anything about that.
• If you took the symbol and made it into the inequality A ≤ B, in this case it would mean that A is smaller or equal to B.
• I'm being very picky right know but if your only eating 1500 calories then you ether have to be a 7 year old or be a professional athlete , speaking that he already has a job at khan academy and is defiantly not a 7 year old you probably only want to do this for a day or two diet .
• ≥o≤ It makes 99.9% sense
• How do we graph the inequalities on a number line?
• If it is less than you draw a line from the number to the left and if it is greater than you draw a line from the number to the right.
(1 vote)
• this was a good thing i learn

## Video transcript

I'm starting to take a little bit more care of my health, and I start counting my actual calories. And let's say C is equal to the number of calories I eat in a given day. And I want to lose some weight. So, in particular, I want to eat less than 1,500 calories in a day. So how can I express that as an inequality? Well, I want the number of calories in a day to be less than-- and remember, the less than symbol, I make it point to the smaller thing. So I want the calories to be less than 1,500. So this is one way of expressing it. I say, look, the number of calories that I consume in a day need to be less than 1,500. Now, one thing to keep in mind when I write that is obviously if I eat no calories in a day, or if I eat 100 calories, or if I eat 1,400 calories, or if I eat 1,499 calories for C, those are all legitimate. Those are all less than 1,500. But what about 1,500 calories? Is it true that 1,500 is less than 1,500? No. 1,500 is equal to 1,500. So this is not a true statement. But what if I want to eat up to and including 1,500 calories? I want to make sure that I get every calorie in there. How can I express that? How can express that I can eat less than or equal to 1,500 calories, so I can eat up to and including 1,500 calories? Right now, this is only up to but not including 1,500. How could I express that? Well, the way I would do that is to throw this little line under the less than sign. Now, this is not just less than. This is less than or equal to. So this symbol right over here, this is saying that C is less than or equal to 1,500 calories. So now 1,500 would be a completely legitimate C, a completely legitimate number of calories to have in a day. And if we wanted to visualize this on a number line, the way we would think about it, let's say that this right over here is our number line. I'm not going to count all the way from 0 to 1,500, but let's imagine that this right over here is 0. Let's say this over here is 1,500. How would we display less than or equal to 1,500 a number line? Well, we would say, look, we could be 1,500, so we'll put a little solid circle right over there. And then we can be less than it, so then we would color in everything less than 1,500, and say, look, anything less than or equal to 1,500 is legitimate. And you might say, hey, but what about the situation where it wasn't less than or equal? What about the situation if it was just less than? So let me draw that, too. So going back to where we started, if I were to say C is less than 1,500, the way we would depict that on a number line is-- let's say this is 0, this is 1,500, we want to make it very clear that we're not including the value 1,500. So we would put an open circle around it. Notice, if we're including 1,500, we fill in the circle. If we're not including 1,500, so we're only less than, we were very explicit that we don't color in the circle. But then we show that, look, we can do everything below that. Now, you're probably saying, OK, Sal, you did less than, you did less than or equal, what if you wanted to do it the other way around? What if you wanted to do greater than and greater than or equal? Well, let's think about that for a second. Let's say that I'm also trying to increase the amount of water I intake. And so let's define some variable. Let's say W is equal to the number of ounces of water I consume per day. And I've read that I should have at least-- let me throw out a number-- 64 ounces of water per day. There's one way I could think about, where I always want to drink more than 64 ounces, so that would be W is greater than 64. W here is the thing that I want to be bigger, so the opening is to the W. W is greater than 64 ounces. How would I depict that? Well, let me do my number line right over here. Let's say that this is 0. This is 64. If I wanted to make strictly greater than, so in this situation it's not cool if I just drink exactly 64. That 64 is not greater than 64. I have to drink 64.01 ounces or 0.00001 ounces. It has to be something that is greater than 64. So I'm not going to include 64, but anything greater than that is completely cool. Now, what if I want to loosen things a little bit? It's OK if I drink exactly 64 ounces or more. Well, then I could write W is greater than or equal to 64. And the way that I would be depict that on the number line-- and obviously, I'm not showing all the numbers in between-- let's say this is 0, and then we go all the way up to 64. Well, now it's OK if I drink exactly 64 ounces, so I'm going to fill in the circle now. Here I opened it because 64 was not a cool number. Now, 64 is completely OK. I can drink exactly 64 ounces of water in the day or more, and then I just go up the number line just like that.