Interval Notation
Open Intervals: Less than or greater than but not equal to (<, >)
If there are two real numbers a and b, then a < x < b means that the numbers represented by x are
between a and b, where a is smaller than b, but the values of x cannot equal a or b.
On the number line, there are open circles at a and b and the line is shaded between them.
Interval notation uses parentheses ( ) to show the open circles; the numbers in the parentheses represent
the ends of the shaded area.
a<x<b
⇒
Interval notation: (a, b)
O
a
O
b
x < a means that the numbers represented by x are less than but not equal to a. This interval includes
numbers from negative infinity (− ∞ ) up to but not including a.
On the number line, there is an open circle at a and the number line is shaded to the left.
Interval notation uses parentheses before − ∞ and after the number a to show the open circle.
x<a
⇒
Interval notation: (− ∞, a )
O
a
x > b means that the numbers represented by x are greater than but not equal to b. This interval
includes numbers from b (but not including b) through positive infinity (∞ ) .
On the number line, there is an open circle at b and the number line is shaded to the right.
Interval notation uses parentheses before the number b to show the open circle and to the right of ∞ .
x>b ⇒
Ex:
Interval notation: (b, ∞)
O
b
x<5 ⇒
Interval notation: (− ∞,5 )
x > –3 ⇒
Interval notation: (–3, ∞)
O
–3
–7 < x < 1
⇒
O
–7
Interval notation: (–7, 1)
O
5
O
1
Closed Intervals: Less than or equal to, or greater than or equal to (≤, ≥)
If there are two real numbers a and b, then a ≤ x ≤ b means that the numbers represented by x are
between a and b, where a is smaller than b, and the values of x can equal a or b.
On the number line, there are closed circles at a and b and the line is shaded between them.
Interval notation uses brackets [ ] to show the closed circles; the numbers in the parentheses represent
the ends of the shaded area.
a≤x≤b
⇒
Interval notation: [a, b]
a
b
x ≤ a means that the numbers represented by x are less than or equal to a. This interval includes
numbers from negative infinity (− ∞ ) up to and including a.
On the number line, there is a closed circle at a and the number line is shaded to the left.
Interval notation uses a parenthesis before − ∞ and a bracket after the number a to show the closed
circle.
x≤a
⇒
Interval notation: (− ∞, a ]
a
x ≥ b means that the numbers represented by x are greater than or equal to b. This interval includes
numbers from b (and including b) through positive infinity (∞ ) .
On the number line, there is a closed circle at b and the number line is shaded to the right.
Interval notation uses a bracket before the number b to show the closed circle and a parenthesis to the
right of ∞ .
x≥b ⇒
Interval notation: [b, ∞)
b
Ex:
x ≥ 10 ⇒
–5 ≤ x ≤ –3
Interval notation: [10, ∞ )
⇒
10
Interval notation: [–5, –3]
–5
x≤4 ⇒
–3
Interval notation: (− ∞,4]
4
Half-open, half-closed intervals: a combination of parentheses and brackets
a<x≤b
⇒
{
a
(a, b]
a≤x<b
b
⇒
[a, b)
{
b
a
Ex:
2≤x<7
2<x≤7
⇒
⇒
Interval notation: [2, 7)
Interval notation: (2, 7]
2
O
7
O
2
7
Summary of Interval Notation
An interval on the real number line is a set of points on the line less than or less than or equal to some
value, greater than or greater than or equal to some value, or between two numbers, either including or
not including the endpoints. We can indicate intervals by writing an inequality, graphing them on the
real number line, or by using interval notation.
Values of x
all x on the line
Graph of Interval
Interval Notation
( −∞ , ∞ )
Open
x<3
less than 3, not
including 3
x<3
less than 3 or
equal to 3
x>3
greater than 3,
not including 3
(- ∞ , 3)
Open
3
(- ∞ , 3]
Half Open
3
(3, ∞ )
Open
3
x>3
greater than 3 or
equal to 3
[3, ∞ )
Half Open
3
3<x<5
between 3 and 5,
not including 3
and 5
3< x < 5
between 3 and 5,
including 3 but
not including 5
3<x<5
between 3 and 5
including 5 but
not including 3
3<x<5
between 3 and 5
including both 3
and 5
(3,5)
Open
3
5
[3,5)
Half Open
3
5
(3,5]
Half Open
3
5
[3,5]
Closed
3
5