Plotting inequalities

I'm starting to take a little bit more care of my help and 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 number of calories in a day and I want to lose some weight so in particular I want to eat less than 1500 calories in a day so how can i express that as an inequality well I want the number of calories in a day I want the number of calories in a day to be less than and remember the less than symbol I make the smaller thing on I make this I make it point to the smaller thing so I want the calories to be less than 1500 1500 so this is one way of expressing I say look the number of calories that I consume in a day need to be less than 1500 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 a hundred calories or if I ate 1400 calories or if I ate 1499 calories for C those are all legitimate those are all less than 1500 but what about 1500 calories is it true is it true that 1500 1500 is less than 1500 No 1500 is equal to 1500 so this is not a true statement but what if I want to eat up to an including 1500 calories I want to make sure that I get every calorie in there how can i express that how can i express that I can eat less than or equal to 1500 calories so I can eat up to and including 1500 calories right now this is only up to but not including 1500 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 is less than less than or equal to equal to 2:15 1500 calories so now 1500 would be a completely legitimate see 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 1500 but let's imagine that this right over here is 0 let's say this over here is 1500 how would we display less than or equal to 1500 on a number line well we would say look we could be 1500 so we'll put a little a little solid circle right over there and then we can be less than it so then we would then we would color in everything less than everything less than 1500 and say look anything anything less than or equal to 1500 is legitimate now 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 if going back to what we were we started if I were to say C is less than 1500 the way we would depict that on a number line is so let's say this is 0 this is 1500 we want to make it very clear that we're not including the value 1500 so we would put an open circle around it notice if we're including 1500 we fill in the circle if we're not including 1500 so we're only less than we we make we were very explicit that we we don't color in the circle but then we show that look we can do everything we can do everything below that now you're probably saying okay Sal you did less than you did less than or equal what if you want to do 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 water I consume per day consume per day and I've read that I should have I should have at least let me throw out a number 64 ounces of water per day so 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 so I always want W here's the thing that I want to be bigger so the opening is to the W W is greater than 64 ounces so 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 60 4.01 ounces or zero zero zero zero one ounce it has to be something that is greater than 64 so I'm not going to include 64 but anything greater than that anything greater than that is completely cool now what if I I want to loosen things a little bit it's like okay it's okay if I drink exactly 64 ounces or more well then then I could write W is greater than or equal to 64 and the way that I would be picked that on the number line the way I would depict that on the number line and obviously I'm not showing all the other numbers in between but let's say this is zero and then we go all the way up to 64 well now it's okay if I drink six 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 okay 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