Now do Exercises 23–34
U
5
V
Intervals of Real Numbers
Retailers often have a sale for a certain
interval
of time. Between 6
A
.
M
. and 8
A
.
M
.
you get a 20% discount. A
bounded
or finite interval of real numbers is the set of real
numbers that are between two real numbers, which are called the
endpoints
of the
interval. The endpoints may or may not belong to an interval.
Interval notation
is
used to represent intervals of real numbers. In interval notation, parentheses are used
to indicate that the endpoints do not belong to the interval and brackets indicate that
the endpoints do belong to the interval. The following box shows the four types of
finite intervals for two real numbers
a
and
b
, where
a
is less than
b
.
Finite Intervals
Verbal Description
Interval Notation
Graph
The set of real numbers
(
a
,
b
)
between
a
and
b
The set of real numbers
[
a
,
b
]
between
a
and
b
inclusive
The set of real numbers greater
(
a
,
b
]
than
a
and less than or equal to
b
The set of real numbers greater
[
a
,
b
)
than or equal to
a
and less than
b
a
b
a
b
a
b
a
b
0
1
2
3
4
5
6
The interval [2, 5)
0
1
2
3
4
5
6
Figure 1.11
Note how the parentheses and brackets are used on the graph and in the interval nota-
tion. It is also common to draw the graph of an interval of real numbers using an open cir-
cle for an endpoint that does not belong to the interval and a closed circle for an endpoint
that belongs to the interval. For example, see the graphs of the interval [2, 5) in Fig. 1.11.
In this text, graphs of intervals will be drawn with parentheses and brackets so that they
agree with interval notation.

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1-7
1.1
The Real Numbers
7
E X A M P L E
4
Interval notation for finite intervals
Write the interval notation for each interval of real numbers and graph the interval.
a)
The set of real numbers greater than 3 and less than or equal to 5
b)
The set of real numbers between 0 and 4 inclusive
c)
The set of real numbers greater than or equal to
1 and less than 4
d)
The set of real numbers between
2 and
1
Solution
a)
The set of real numbers greater than 3 and less than or equal to 5 is written in
interval notation as (3, 5] and graphed in Fig. 1.12.
1
2
3
4
5
6
The interval (3, 5]
Figure 1.12
1
1
0
2
3
4
5
The interval [0, 4]
Figure 1.13
Some sales never end. After 8
A
.
M
. all merchandise is 10% off. An
unbounded
or
infinite interval
of real numbers is missing at least one endpoint. It may extend infi-
nitely far to the right or left on the number line. In this case the infinity symbol
is
used as an endpoint in the interval notation. Note that parentheses are always used
next to
or
in interval notation, because
is not a number. It is just used to indi-
cate that there is no end to the interval. The following box shows the five types of infi-
nite intervals for a real number
a
.
c)
The set of real numbers greater than or equal to
1 and less than 4 is written in
interval notation as [
1, 4) and graphed in Fig. 1.14.