INTERVAL
NOTATION
Notes, Examples, and Two Practice Sets
Janet Knox
Helpful
Charts
Parentheses v. Brackets
If an inequality has a
<
OR
>
then the number line will
have an OPEN circle
and the interval notation
will have a
(
OR
)
If an inequality has a
≤
OR
≥
then the number line will
have an SOLID dot
and the interval notation
will have a
[
OR
]
EXAMPLES
Interval Notation
Let’s see an example:
x≤3
+∞
–∞
3
(– ∞, 3]
Negative infinity
always has a
parenthesis.
Another example:
x<3
+∞
–∞
3
(– ∞, 3)
Negative infinity
always has a
parenthesis.
Example:
x≥3
+∞
–∞
3
[3, +∞)
Positive infinity
always has a
parenthesis.
Example:
x>3
+∞
–∞
3
(3, +∞)
Positive infinity
always has a
parenthesis.
NOTES
Interval Notation
A solution set for x
can be expressed
three ways:
1)
Inequality
2) Number Line
3) Interval Notation
Interval Notation
is the NEW
Method
To covert a shaded number line into
INTERVAL NOTATION,
read the number line from
LEFT TO RIGHT.
Example:
–∞
+∞
-3
-2
-1
[-2, +∞)
The shading starts at and INCLUDES -2.
The shading ends at positive infinity.
Example:
–∞
+∞
-3
-2
-1
(– ∞, -2)
The shading starts at negative infinity.
The shading ends at -2,
but does NOT INCLUDE -2.
Example:
–∞
+∞
-3
-2
-1
(-3, -1]
The shading starts at -3,
but does NOT INCLUDE -3.
The shading ends at and INCLUDES -1.
Union of Sets
Example:
The number line represents
the union of two sets.
Write the notation for each interval.
Unite the intervals with a U symbol.
(-∞, -3) U [0, +∞)
How would you write
the interval notation for the
set of all real numbers?
The shaded number line includes every real
number from negative infinity up to positive
infinity.
(-∞, +∞)
PRACTICE
Set One
PRACTICE PROBLEMS, SET 1
Write in interval notation:
ANSWERS
(-∞, 3)
(-1, 3)
(-∞, -2] U [1, 3)
(-∞, 1) U (1, ∞)
[2]
(-∞, -3] U (-1, 1] U (3, ∞)
[-3, -1] U (1, ∞)
PRACTICE
Set Two
PRACTICE PROBLEMS, SET 2
Draw the number line and write the interval
notation for #1 – 4.
1.)
x≤5
2.)
x>0
3.)
–1 ≤x<2
4.)
x > 2 or x ≤ – 1
5.) A certain type of spider cannot grow
beyond six inches. Write an inequality for the
length of this bug and state your final answer
in interval notation.
Solutions
1.)
x≤5
–∞
+∞
4
5
6
(-∞, 5]
2.)
x>0
–∞
+∞
-1
0
1
(0, ∞)
Solutions, Continued:
3.)
-1 ≤ x < 2
–∞
+∞
-1
2
[-1, 2)
4.)
x > 2 or x ≤ – 1
(-∞, -1] U (2, +∞)
Solutions, Continued:
5.) A certain type of spider cannot grow
beyond six inches. Write an inequality for the
length of this bug and state your final answer
in interval notation.
The spider cannot have a negative
or zero length.
The length is between 0 and 6 inches.
The spider can be six inches, so 6 is
included in the solution set.
0< x≤6
(0, 6]
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