1. [ Intervals ]

2. We need to REVIEW certain things…
But FIRST …
We need to REVIEW certain things…

3. Important Symbols x < 9 x > 9 x ≤ 9 x ≥ 9
This means that x is less than
but not equal to 9
This means that x is greater than
but not equal to 9
x ≤ 9
x ≥ 9
This means that x is
less than or equal to 9
This means that x is
greater than or equal to 9

4. Definition: Back to Intervals…
Any set of real numbers that is represented on the number line by a segment is called an INTERVAL

5. Some Important Notation
Brackets:
Identified by the symbols [
and ]

6. [4,9]
This is a closed bracket. Everything between the brackets is in the interval.
4, 5, 6, 7, 8, 9
]2,7[
This is an open bracket. Everything between the brackets is in the interval EXCEPT the end points.
3, 4, 5, 6

7. [7,10[
This is closed on the left and open on the right. Everything between the brackets EXCEPT the right end point is in the interval.
7, 8, 9
]12,14]
This is open on the left and closed on the right. Everything between the brackets EXCEPT the left end point is in the interval.
13, 14

8. Which numbers are part of the following intervals:
Some Examples
Which numbers are part of the following intervals:
1. [7,13[
2. [4,8]
7, 8, 9, 10, 11, 12
4, 5, 6, 7, 8
4. ]24,28]
3. ]4,6[ 5
25, 26, 27, 28

9. List four different ways to write the interval containing the numbers 17, 18, 19 and 20
[17,20]
]16,21[
[17,21[
]16,20]

10. Graphic Representation
Some More Important Symbols:
● is a closed circle, means included
○ is an open circle, means excluded

11. Example
The interval would be represented by:
]1,6]

12. What about infinity?
[1,+∞[

13. is made up of real numbers
Set-Builder Notation
{ x є R | a ≤ x ≤ b }
This left side says
that x (our interval)
is made up of real numbers
This right side says
that x (our interval)
has end points a and b

14. How to FIND the interval
{ x є R | 1 ≤ x < 4 }
Step 1: Write the end points
1,4
Step 2: Place the brackets – LOOK AT EACH SIDE!
Left Side: 1 ≤ x
Can x be 1?
YES!
So we have a closed bracket: [1,4
Right Side: x < 4
Can x be 4?
NO!
So we have an open bracket: [1,4[

15. Another Example { x є R | x < 10 } -∞, 10
Step 1: Write the end points
-∞, 10
Step 2: Place the brackets – LOOK AT EACH SIDE!
Left Side: There is none! The interval goes to -∞!!
So we have a closed or open bracket: [-∞, 10 or ]-∞, 10
Right Side: x < 10
Can x be 10?
NO!
So we have an open bracket: [-∞, 10[ or ]-∞, 10[

16. Helpful Trick! Open bracket = Open circle = < or >
Closed bracket = Closed circle = ≤ or ≥
Open bracket = Open circle = < or >