[ Silence]
>>In this video I'm going to go
over what is called interval notation.
It's a different way of writing an inequality
and graphing an inequality.
So let's start off with this problem, X is greater than
or equal to 1, this is more standard notation.
And if I were to graph that it says X is greater than or equal
to 1 so that means it's equal to 1 so I put a dot on 1
and then it says it also can be greater
than so I draw an arrow to the right.
And that's one way of graphing X is greater than or equal to 1.
Now there's a completely different notation,
it looks similar, where you show, using either parenthesis
or brackets, the smallest number
and the largest number in the inequality.
So in this particular problem,
the smallest number is actually 1 and if it actually is equal
to 1, what we do is we start off with a bracket.
So you're going to either use a parenthesis or a bracket.
I'll explain the parenthesis in a minute.
A bracket means this is the very smallest number
of the solution, 1.
And then notice it goes on forever though.
There is no highest number.
But if you go infinitely to the right you have to imagine,
well what number could that be?
There is none.
We'll just say infinity and we use open parenthesis
to show it doesn't really land on a specific number.
It keeps on going forever and ever to infinity
and infinity looks like a sideways 8.
Okay, now how would I graph this using interval notation?
Well if it goes infinitely to the right
and the smallest number is 1, what we'll do is
at 1 we'll put the bracket going the same way
so we have a bracket right at 1, showing it's opening
to the right and then we do the exact same thing
with an arrow going to the right.
So this is a way of showing X is greater than or equal
to 1 using interval notation.
Alright now what if instead of it being greater than or equal
to 1 it was just greater than something?
Let's do one like that.
Let's say the problem was X was greater than negative 2.
Alright so what's the smallest number it can be?
Well you know, it's greater than negative 2,
there's no smallest number just to the right, right?
So this one's a little bit tricky.
But when you were using the older notation,
what we would do is put a circle on negative 2
to show it doesn't exactly touch negative 2
and then we would just draw an arrow going to the right.
Okay so the difference is instead
of a dot we'll put a circle.
Circle means negative 2 is not part of the solution.
Alright now for interval notation,
what's the smallest number?
Well there is no smallest number.
If there is no smallest number then we use a parenthesis.
So this means it goes almost to negative 2, okay, to the left,
of course, it hits almost negative 2.
And then I go to the right
and again I'm going all the way forever to the right,
I'm gonna use infinity and a parenthesis.
Anytime you're dealing with infinity all the way
to the right or negative infinity all the way
to the left, we're going to use a parenthesis next to infinity.
Also in interval notation you put the smaller number
on the left and the larger number or infinity on the right.
So how would we graph this?
Instead of using the bracket on negative 2,
we're gonna use the parenthesis on negative 2
and then we'll go to the right.
So it looks similar, right,
instead of being an open circle we'll use the parenthesis.
It sort of matches what it looks like over here.
Now let's see, by the way, we might want
to just make this the same color, matching it up.
Alright, so now let's do one where it's like less than.
It's going to be exactly the same except the arrow will be
going to the left.
So here's another example, say X is less than 3,
alright now notice 3 is not a solution
so in my usual notation I'm going to use a circle
because it doesn't exactly touch 3
and then I want all the numbers to the left of 3, right,
X is less than 3, so I need to go to the left
with my arrow okay so this is the usual notation most students
first learn.
Now what we want to do is how am I gonna write this using
interval notation?
Alright so I want numbers to the left of 3
so look at the picture here.
It's going to the left forever
so what's the very smallest number?
Well there is no smallest number so we call
that negative infinity
and so we're gonna have to use a parenthesis.
Negative infinity and it's going to go all the way until 3
but not exactly touch it goes up to 3
and since it doesn't touch it, we put a parenthesis around it.
Alright so now let's do the matching interval notation.
Instead of a circle it's going,
it's going to be a parenthesis opening the same way, right,
where it is over here with a 3
and we're going to the left forever.
So there's the interval notation way
of righting the inequality in the graph.
Now notice this looks like an ordered pair, it is not.
It's sort of like you have to take it in context.
Just like if you look at a certain word, like this word,
and I ask you how would you say that?
How would you pronounce it?
One person might say red and another person might say read.
It's all in context of the sentence it's in.
It's the same thing here.
You have to take it into context.
We're talking about a solution in interval notation
in equalities so this is not an ordered pair.
Also the infinity is a clue that something's strange.
But we are gonna do some other problems
where they'll be two numbers and it really does look
like an ordered pair and it's not.
So here's an example like that, what if I had that X was
between negative 1 and positive 3?
And in fact, this is read, negative 1 is less
than X is less than or equal to 3.
So if we want to graph this the usual way,
alright it says X is really sandwiched between negative 1
and 3 but it doesn't exactly equal negative 1.
It's a little bit bigger than negative 1 so I put the circle
on negative 1 but it actually goes all the way
up to positive 3.
So we fill in between those numbers.
Those are the numbers that are bigger than negative 1
and go all the way up to and including 3.
So now how are we gonna write that in interval notation?
Okay well if it's sandwiched between two numbers,
actually it's kind of easy.
We're gonna write those two numbers in order,
smallest number on the left
and if it does not have an equal sign there's gonna be a
parenthesis and if it does have an equal sign it's going
to have brackets.
So that's how you would write negative 1 is less
than X is less than or equal to 3, using the interval notation.
So let's see what that looks like on a graph.
It matches up really quite nicely.
Alright so we have the same type of thing going on,
negative 1 we have the open parenthesis
and at 3 you have the bracket and then you fill in
[noise]
to get your solution.
[noise] Alright so that's basically how you would use
interval notation and what's nice is it's a little bit
shorter to write, you know to write the negative 13 this way
as opposed to writing negative 1 is less
than X is less than or equal to 3.
So this is just another notation that you commonly use
when you're trying to write a solution set with inequalities.
Keep in mind that if this was negative 1 is less
than X is less than 3, you would have had a parenthesis instead
of a bracket, next to the 3 and that's what I'm talking
about it really looked just like an ordered pair, negative 13
but it's not because it's in context.
We're talking about how to write a solution to an inequality.
So you just have to keep in mind what you're writing.
Alright, one last thing, what
if you wanted the entire number line to be your answer?
That would be negative infinity to positive infinity
and you always use parenthesis.
And the way this would look on a graph,
I won't say that's negative 5 and 5 for instance,
it's just the whole thing, the whole graph.
So if you've got the whole graph, the whole number line,
it's just between negative infinity and positive infinity.
Let's do this in general.
We'll assume A is less than B. Let's say we were given these
inequalities, X is less than A. This is what the graph would
look like interval notation.
This is what the interval notation would look
like for these four possibilities.
We have these other possibilities
where X is squished between two numbers.
Let's see, I forgot to put that there.
And so that's it in general.
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