Ordering expressions

- Let's get some practice understanding the 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 and q on the number line. So just to get our bearings, let's see, three hash marks to the left of zero is negative three. So each hash mark we must be going down by one. So this must be negative one, negative two, and 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 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 they're both negative or both positive or one's negative and one's positive. Just the fact that we know that 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, n is negative. If you subtract a negative, you're going to essentially add a positive. So 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 had to compare it versus q, we would know that it's 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 one and negative two. It looks like it's approximately negative 1.8. We don't know for sure but if we just eyeball it, this thing 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 the larger number. So this thing right over here is going to be negative. So the largest of these values 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 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, and this is the largest, or the greatest I should say maybe. Maybe let me call this the least. That might be better because sometimes when people say small and large they're referring to absolute value. But let's say this is the least and this is the greatest. So if we wanted to order them we would go n minus q, and then if you're doing this on Khan Academy exercise, you can actually click on these and move them around, but 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 positive 2.6. So this is the greatest. And so let's do the next question. Order the, whoops, order the following expressions by their values from least to greatest. So once again the 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 1/2 right over here. And we want to 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 minus zero. Right that's the same thing as a. So let's see. In all these we have an a and we're subtracting something. Where in here we're subtracting zero. I guess we're subtracting nothing. Here you're subtracting b. And here you're subtracting 0.5. So in general the more you subtract, the more that you subtract, the smaller it's going to be. So let's see. 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 a, you're going to get a lower value than if you subtract 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 or the greatest value. So a minus 0.5. We're subtracting a positive number there. We're subtracting 0.5. Followed by a where 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 going to get a value that is greater than a. Let me make it very clear. This value right over here 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 negative .2, so approximately negative 0.2. Remember this is negative half. This is negative .5. a looks like it is approximately, I don't know, negative .7, negative 0.7. And so you could actually do it with the actual values if you like, but you would get the same result.