In this exercise,
we'll practice expressing intervals
using inequality notation,
using a graph,
and also using interval notation.
First we have the inequality
x is greater than negative three.
Notice here, negative three is not in this interval.
So to show negative three is not in the interval,
we would make an open point on negative three
using the Open Dot tool.
And then because x is greater than negative three,
we would graph to the right using the Line Segment tool.
This is the interval where
x is greater than negative three.
Another way to express the same interval
would be to use a rounded parenthesis
on negative three like this
rather than an open point.
This indicates the end point of negative three
is not in the interval.
So this is a second way to graph the exact same interval.
And before we talk about interval notation, notice how
as we move right on the number line,
we would approach positive infinity.
So using interval notation,
we have the interval from negative three on the left
approaching positive infinity on the right.
And because negative three is not in the interval,
we use a rounded parenthesis here.
If it was included, we'd use a square bracket.
And for positive and negative infinity,
we always use a rounded parenthesis.
To enter infinity into our homework system,
you could either enter two lowercase Os on the keyboard
or use the equation tool
and select the infinity symbol.
Next we have x is less than or equal to negative four.
Notice here, negative four is in the interval
because of the equal part.
So to show negative four is in the interval,
make a closed point,
and x equals negative four here,
using the Dot tool.
And then because x is less than or equal to negative four,
we would graph to the left
using the Line Segment tool.
Instead of using a closed point, though,
we can also place a square bracket here,
x equals negative four,
indicating this endpoint is in the interval,
and then graph to the left.
Notice here, to the left
we're approaching negative infinity.
So using interval notation,
on the left we're approaching negative infinity.
The right endpoint is negative four.
Because negative four is in the interval,
we use a square bracket here,
and, again, for infinity or negative infinity
we use a rounded parenthesis.
Now for our next two examples,
we're given the interval notation.
We want to provide the graph
as well as the inequality notation.
So looking at the interval notation,
we have the interval from one to infinity.
Because of the rounded parenthesis here,
one is not included in this interval.
So we'd have an open point on one.
We would graph to the right
approaching positive infinity, or
the alternative notation would be to use
an open parenthesis here
on one and graph to the right.
The inequality notation would be
all the numbers that are greater than positive one,
so we say x is greater than positive one.
Next we have the interval from negative one to three.
Because we have a square bracket on the left,
negative one is in the interval.
And we have a rounded parenthesis on the right,
and therefore three is not in the interval.
So the graph is
we would have a closed point on x equals negative one
to show negative one is in the interval,
we would have an open point on positive three here,
and we graph the interval between these two.
Or we could also have a square bracket on negative one,
a rounded parenthesis on three,
and graph between.
The inequality is going to be a compound inequality
where x is less than three
and also greater than or equal to negative one.
So greater than or equal to negative one.
Remember, we could read this compound inequality two ways.
Starting in the middle,
we get x is less than three in this direction
and also x is greater than or equal to negative one.
I wrote this up here because
to enter this into our homework system we enter
negative one less than or equal to
x less than three.
So less than or equal to symbol must be entered
this way here.
And it's always important to use the Preview button here
to see how the homework system interprets your entry.
Now for our last two examples,
we're given the graph.
We want to express the interval notation
as well as the inequality notation.
Let's start with the inequality.
Notice here, one is in the interval,
so we say x is greater than or equal to positive one.
So this is the correct inequality notation,
but we enter x greater than equals one.
Click Preview, and it will show us this notation.
And using interval notation,
we have the interval from one on the left
approaching positive infinity on the right.
The interval includes one,
so we use a square bracket here
and a rounded parenthesis for infinity.
And for our last example
we have the interval from negative three to two.
The interval includes negative three
and does not include positive two.
So we can say that x is less than two
and greater than or equal to negative three.
So starting with x here, we'd say
greater than or equal to negative three.
So x is between negative three and two,
including negative three
and not including two.
The way we enter this in, though, is going to be
negative three less than equals x less than 2.
Click Preview, and we'll see this notation.
And finally for the interval notation,
we got the interval from negative three to two.
Because we include negative three
we have a square bracket.
Because we don't include two
we have a rounded parenthesis here.
And before we go, another way to graph this interval here
would be to use a square bracket on the number line
and graph to the right,
and for this interval we'd have a bracket
on negative three,
a parenthesis on positive two,
and we graph between the values.
Notice here, the scaling of the number lines are different.
I hope you found this helpful.