Name: ____________________________________
Date: _________________
Interval Notation and Infinite Sets
Algebra 1
Sets of numbers that comprise intervals along a number line are of particular interest in mathematics.
We have seen how to represent these intervals using set builder notation. Now we will introduce an
alternative called interval notation. In this notation, [ ] are used for closed circles and ( ) are used
for open circles and the number line is omitted. The interval 3  x  2 would be written as  3, 2 .
Exercise #1: Sets representing intervals are shown on the number lines below. Represent each set
using set builder notation and interval notation.
Graphed Interval
-10 -8 -6 -4 -2
0
2 4
6 8 10
-10 -8 -6 -4 -2
0
2 4
-10 -8 -6 -4 -2
0
2 4
6 8 10
-10 -8 -6 -4 -2
0
2 4
6 8 10
-10 -8 -6 -4 -2
0
2 4
6 8 10
-10 -8 -6 -4 -2
0
2 4
6 8 10
-10 -8 -6 -4 -2
0
2 4
6 8 10
I -10 -8 -6 -4 -2
0
2 4
6 8 10
6 8 10
Algebra 1, Unit #11 – Sets and Counting – L2
The Arlington Algebra Project, Lagrangeville, NY 12540
Set Builder Notation
Interval Notation
Interval notation can be somewhat confusing because it closely resembles the way we specify a
coordinate point  x, y  . It will always be clear from the context of the problem whether you are
dealing with a coordinate point or an interval. Thus, always read questions carefully to understand
what is being asked.
Exercise #2: The set given in set builder notation as x : 3  x  7 can also be expressed as which of
the following?
(1)  3, 7 
(3) [3, 7]
(2)  3, 7
(4)  3, 7 
Exercise #3: The solution set to the inequality 2x  5  15 can be expressed as
(1)  ,  5
(3)  5,  
(2)  ,  5
(4)  5,  
There is some additional terminology associated with intervals along a number line. If an interval
contains all of its endpoints we call it inclusive and closed. If an interval lacks both of its endpoints
we call it exclusive and open. If an interval contains one of its endpoints but not the other, we call it
half-closed (or half-open).
Exercise #4: Which of the following inequalities represents the set of all real numbers between -8 and
4 inclusive?
(1)  8, 4 
(3)  8, 4
(2)  8, 4
(4)  8, 4
Exercise #5: Which of the following intervals is half-closed?
(1)  5,  
(3)  6,10
(2) 3, 9
(4)  , 2
Algebra 1, Unit #11 – Sets and Counting – L2
The Arlington Algebra Project, Lagrangeville, NY 12540
Name: ____________________________________
Date: _________________
Interval Notation and Infinite Sets
Algebra 1 Homework
Skills
1. Represent each interval graphed below with both set builder and interval notation.
Graphed Interval
Set Builder Notation
-10 -8 -6 -4 -2
0
2 4
6 8 10
-10 -8 -6 -4 -2
0
2 4
-10 -8 -6 -4 -2
0
2 4
6 8 10
-10 -8 -6 -4 -2
0
2 4
6 8 10
-10 -8 -6 -4 -2
0
2 4
6 8 10
-10 -8 -6 -4 -2
0
2 4
6 8 10
-10 -8 -6 -4 -2
0
2 4
6 8 10
-10 -8 -6 -4 -2
0
2 4
6 8 10
-10 -8 -6 -4 -2
0
2 4
6 8 10
6 8 10
Algebra 1, Unit #11 – Sets and Counting – L2
The Arlington Algebra Project, Lagrangeville, NY 12540
Interval Notation
2. The set of all real numbers less than or equal to 5 could be expressed as
(1)  5,  
(3)  , 5
(2) 5,  
(4)  , 5
3. The set x : 10  x  8 can be written in interval notation as
(1)  10, 8
(3)  10, 8
(2)  10, 8
(4)  10, 8
4. The set of all positive real numbers can be expressed as which of the following?
(1) 1,  
(3)  0,  
(2)  0,  
(4)  , 0
5. Which of the following represents a closed interval?
(1)  5, 4
(3)  3, 7 
(2) 7,12
(4)  6, 2
6. Which of the following intervals represents all numbers between 5 and 10 exclusive?
(1) 5,10
(3)  5,10
(2)  5,10
(4) 5,10
7. The solution set to the inequality 35  4x  11 can be written as
(1)  , 6
(3)  6,  
(2)  6,  
(4)  , 6
Algebra 1, Unit #11 – Sets and Counting – L2
The Arlington Algebra Project, Lagrangeville, NY 12540