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## Adding & subtracting negative numbers: variables

## Video transcript

- 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.