In this exercise, we are given several inequalities
then asked to graph the interval
and then express the interval using interval notation.
Looking at the first inequality, we have X less than five.
Because inequality is less than, not less than
or equal to, five is not in the interval
which means we begin by graphing or plotting
an open point or open circle on five.
The open point indicates this value is not in the interval.
If it said less than or equal to, then we'd have
a closed point, indicating the value is in the interval.
And then because the inequality is X less than five,
values less than five are to the left.
We now graph to the left.
Notice how the values approach negative infinity.
This is the graph for the inequality X less than five.
Now let's express the interval using interval notation.
Because the values approach negative infinity
to the left and positive five to the right,
we begin with negative infinity comma five
and now need to decide whether to the right or left
we have a square bracket round parenthesis.
For positive or negative infinity,
we always have a rounded parenthesis
and then because interval does not include five,
we also have a rounded parenthesis to the right of five.
If it included five, we would have a square bracket
to the right of five.
For B, we have X greater than negative two.
Because the inequality is greater than not greater than
or equal two, negative two is not in the interval.
We begin by making an open point or open circle
on negative two.
This indicates negative two is not in the interval
and now because the inequality is X greater
than negative two, values are greater to negative two
to the right.
We graph to the right approaching positive infinity.
And now for the interval notation,
the values here are approaching negative two to the left
and positive infinity to the right.
We begin with negative two comma infinity.
Because negative two is not in the interval,
we have a rounded parenthesis to the left
and for positive and negative infinity,
we always have a rounded parenthesis.
This is the interval using interval notation.
Next, we have X less than or equal to negative one.
Because the inequality is less than or equal to negative one
negative one is in the interval,
we make a close point on negative one at this time
because negative one is in the interval.
The interval is X less than or equal to negative one.
The values are less than negative one to the left.
We graph to the left, approaching negative infinity,
which means for interval notation,
we begin with negative infinity comma negative one.
For negative infinity, we always have a rounded parenthesis
because the interval includes negative one,
we have a square bracket to the right of negative one.
So we can say the interval is closed on negative one for C
and for B, we can say the interval is open on negative two.
D is a little bit different.
D, we have X equals three
because the interval includes three,
we make a closed point on three
and this is the only number or only value in the interval,
so we are done with the graph.
It seems a little strange to express this
using interval notation,
but the interval begins and ends with three,
so one way to express this using interval notation
would be three comma three.
And because three is in the interval,
we would have a square bracket to the left of three
and to the right of three.
Another way to express this
using set notation would be to put a three in braces,
which would look like this.
For E, we have X greater than or equal to two.
The interval includes two.
We make a closed point on two.
We have X greater than or equal to two.
Values greater than two are to the right.
We graph to the right approaching positive infinity.
Using interval notation, the interval goes from two
and approaches infinity and includes two.
We have a square bracket to the left of two.
Positive infinity goes to the right forever
and therefore, it's never going to include infinity.
We make a rounded parenthesis to the right of infinity.
The last two are a comp of inequalities.
There's a couple ways to read this.
We can read it from left to right as negative three
less than or equal to X less than four,
but we can also start with the variable
and read this from right to left
as well as left to right.
We could say X is greater than or equal to negative three
and X is less than four.
Either way is correct.
Looking at the inequality symbol though,
notice how negative three is in the interval
and positive four is not, so we make a closed point
on negative three and an open point on positive four.
Again, we can read this as X is greater than or equal to
negative three, so we would graph to the right
of negative three.
But X is also less than four, so we graph to the left
of four, which means this compound inequality
is the interval between these two values,
including negative three and not including four.
To express this using interval notation,
we begin at negative three and we end at four.
The interval includes negative three.
We make a square bracket to the left of negative three.
It does not include four.
We use a rounded parenthesis to the right of four.
And finally for G, we can read this as negative eight
less than X, less than or equal to zero.
Or we can read this as X is greater than negative eight
and X is less than or equal to zero.
The interval does not include negative eight
because we read this X greater than negative eight.
We graph an open point on negative eight
and X is also less than or equal to zero.
Zero is in the interval, so we graph a closed point on zero
and we graph all the values between at negative eight
and zero.
This is the graph of the interval.
The interval is between negative eight and zero.
Negative eight is not in the interval.
We have a round parenthesis to the left of negative eight.
The interval does include zero and therefore,
we have a square bracket to the right of zero.
So again, if the end point is included,
we use a square bracket.
If it's not included or it's plus or minus infinity,
we always use a rounded parenthesis.
I hope you found this helpful.