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let's get some practice understanding variables and the negative numbers that they might represent or the positive numbers so we're told to order the following expressions by their values from least to greatest and they've given us these three expressions Q minus n n and n minus Q and then they plot n NQ on the number line so just to get our bearings let's see this three hash marks to the left of zero is negative three so each hash mark must we must be going down by one so this must be negative one negative two and then this is negative three and then so as we go to the right each hash mark must increase by one so zero one two and then three and then this just helps us get a little bit more bearings but let's just think about each of these expressions so this first one is Q minus N and Q is to the right of n on the number line we know that Q is greater than n so if Q is greater than n and you're subtracting n from Q it actually doesn't matter if Q if they're both negative or both positive ones negative ones positive just the fact just the fact that we know that one the Q is greater than n that means that Q minus n is going to be positive and if you actually want to look at this particular circumstance Q is positive and is negative if you subtract a negative you're going to essentially add a positive so this is this value right over here not only is it going to be positive it's going to be a positive value greater than Q and if we have to compare it versus Q we would know that it is greater than Q but they don't ask us to do that now we have n n is a negative value it's a negative number and it's a negative number between negative 1 and negative 2 it looks like it's roughly I don't know it looks like it's approximately negative 1.8 we don't know for sure but if we just eyeball it this this thing is is negative and it looks like it's approximately negative 1.8 now what's n minus Q n minus Q so it's going to be the negative of Q minus n so n minus Q we have the smaller number and from that we're subtracting we're subtracting the larger number so this thing right over here is going to be negative so the largest of these Val is definitely going to be Q minus n which is going to be positive and then we have to figure out which is going to be more negative this n value or this n minus Q value well let's think about it a little bit we could just try to approximate what Q is and if we look at it Q looks like it's approximately this looks like roughly 0.8 and this looks like it's approximately we've already said negative 1.8 so if we make those assumptions right over here this thing is going to be approximately approximately negative 1.8 minus 0.8 which is equal to negative 2.6 so when you look at it like this you clearly see that this is going to be more negative than this right over here so this is the smallest this is the smallest and this is the largest or the greatest I should say maybe maybe let me call this the least that might be a better the least because sometimes when people say small and large they're referring to to absolute value but let's say this is the least and this is this is the greatest so if we wanted to order them we would go n minus Q and then and if you're doing this on the Khan Academy exercise you can actually click on these and move them around but if if we can't it will be n minus Q which is the most negative then you have n which is still negative but not as negative this is roughly negative 1.8 this is roughly negative 2.6 and then Q minus n which is going to be roughly roughly positive 2.6 so this is this is the greatest and so let's do the next question let's do the next question order the Phillips order the following expressions by their values from least to greatest so once again kind of same drill although here each hash mark looks like it's a half because it takes two to get to one so this is half this is negative one half right over here and we want to cut compare a minus B to a to a minus 0.5 so all of them were either a you can even think of this as a minuses a-minus zero all right that's the same thing as a so let's see in all of this we have an A and we're subtracting something we're here we're not subtracted we're subtracting zero I guess we're subtracting nothing here is subtracting B and here you're subtracting zero point five so in general the more you subtract the more that you subtract the more that the smaller it's it's going to be so let's see which in which of these cases am i subtracting the most well here I'm subtracting a positive number here I'm subtracting zero here I am subtracting a B is a negative number here I am subtracting a negative number B is negative this is clearly positive so if you subtract a positive number from the city if you're from a you're going to get a lower value than if you've tracked a negative number in fact if you subtract a negative number you're going to add to a you're going to get a number greater than a so the least is when you subtract the largest value so if you are the greatest value so a minus zero point five we're subtracting a we're subtracting a positive number there we're subtracting zero point five followed by a or we're not subtracting anything and then you have a minus B this right over here is going to be the greatest why is this well we know that B is a negative number notice it's below zero right over here so if B is a negative number you subtract a negative number you're actually gonna get a value that is greater than a let me make it very clear this value right over here is going to be is going to be less than a this value over here clearly equals a and this value over here is actually going to be greater than a so now we've ordered it from least to greatest once again if we were doing it on the Khan Academy exercises we would have a little tool where we could click and move these around but this is the least followed by this followed by that now another way that you could do it just like we did in the last example you could try to estimate roughly what these values are B looks like it's I don't know it's not exactly it looks like it's about point negative 0.2 so approximately negative zero point two remember this is negative half this is negative 0.5 a looks like it is approximately a negative and a point Oh point seven negative zero point seven and so you can actually do it with the with the actual with the actual values if you like but you would get the same get the same result