Writing interval notation. Our goal today is to write sets of points using interval
notation. We've already written sets of points by using graphs and inequalities,
but today we want to learn how to do it using something called interval notation.
And once you learn it, I think you'll like it.
Let's go ahead and take a look at the definition of interval notation.
Now this looks a little busy. What I'd like you to do is focus in on the second column,
and the second column is called set notation. The way you read this, you would say x such that a
is less than x and x is less than b. Now again, what I want to focus on is this inequality here,
compound inequality. And you can see that x is less than b and x is greater than a.
The graph of this, from what you're probably used to seeing, would be something like this,
where the value of a is on the left. It's an open circle because of the inequality
symbol. b is on the right. It's also an open circle because of the inequality symbol,
and it's the interval between those, often called an open interval because these are open points.
Now another way to express this same graph, instead of using open points,
you can use parentheses or rounded parentheses.
Those mean exactly the same thing. The nice thing about this is once you have the graph,
it makes an easy transition to interval notation. If you look on the far left, you can communicate
the same information by using this notation here: such that the interval notation a, comma, b
with rounded parentheses. This is the open interval from a to b.
This first column, second column, third column, and fourth column
all communicate the same information in a different way. If you look at the next row here,
the only difference here is this is the closed interval because the values of a
and b are included in the interval. So this graph would look very similar to the first,
except that at a, you'd have a closed point, and at b, you'd have a closed point as well.
So what you can get from this is that closed points are the same thing as a square bracket.
These two graphs are equivalent. Again, once you have this graph, it makes a nice easy transition
to interval notation. The closed interval from a to b can be expressed in this manner:
a, comma, b with both ends as square brackets.
Now these next two I won't graph, but you can see that's just a combination, where
this is what you would call closed, a closed interval at a
and an open at b, and next row open at a and closed at b, just meaning,
for example, in this case it would have an open circle or open point at a or a closed point at b.
Let's skip down to this next example. Again, I want to kind of make the connection between this,
you know, this inequality here, the way you would read this again: x such that x is greater than a.
Let's graph that. We'd have an open point at a, and greater than a would go to the right.
Now if you go to the right of the number line, you are going to approach positive infinity,
and I like to add that in there. Again, the only difference between these two graphs,
this is using an open point, and this is using a rounded parenthesis. However,
once you have the graph, if you take a look at this graph and you jump all the way to the left
to interval notation, notice there's the a, there's the infinity.
Now there's rounded parenthesis a because of the open point, and if you have an
infinity or negative infinity in interval notation, you're always going to use the
rounded parenthesis because technically infinity and negative infinity do not exist.
The only difference between this next row is that it's closed on a, so we have the square bracket.
Let's look at a couple more examples here or definitions.
Notice on this graph, this would correspond to an open point. I won't graph it,
and if you go to the left, you are approaching negative infinity. So as you can see from the
far left, the interval notation negative infinity to b, both with open parentheses.
The last example here, if you're graphing numbers on the number line,
of course, in the direction positive direction, positive infinity, negative direction, negative
infinity, and that's where this interval notation comes from. It's the graph of all real numbers.
Let's take a look at some specific examples. Write the interval notation for each set. This interval
starts at negative one, goes to positive two. It includes negative one, so that means we have to
write or use a square bracket. It does not include two, so it would use a rounded parenthesis.
This next example, as you can see, the graph goes left forever, that approaches negative infinity.
Again, I like to include that because it helps me make the transition to interval notation.
The interval notation for this interval would be from negative infinity
to positive four, open on four, and of course, always open on negative infinity.
A couple more. Now we want to graph and write interval notation for each set. So let's graph
x is less than negative two.
It'd be an open circle, an arrow to the left. That's really all that I need, but I do like
to include again the negative infinity going to the left. So my interval notation would be from
negative infinity to negative two, and it would be open on both. Two more here, and we'll be done.
x such that negative two is less than x and x is less than or equal to three.
I have a compound inequality here. Let's go ahead and graph it.
x is kind of sandwiched between negative two and positive three. It
does not include negative two, but it does include positive three, and the interval is
between those. The interval notation for this, we know, is from negative two to positive three.
It includes positive three, square bracket. It does not include two, rounded parenthesis.
It is important to note that the values in here should occur as they occur on the number line.
The number that's less should be on the left; the number that's greater should be on the right
in all cases. Last example: x such that x is less than negative two or x is greater than one.
x is less than negative two.
x is greater than one.
And just for added meaning, we're approaching positive infinity as we go to the right;
we're approaching negative infinity as we go to the left.
Now, as you can see, there are two distinct intervals here. Therefore, we are going to need
two interval notations to communicate this. The first one would be from negative infinity
to negative two, both with rounded parentheses, and we also need from one to positive infinity.
And we could write the word or in here; we could also use the symbol for union. I hope that helps.
If you need any more adjustments, just let me know!