Intervals

	From 1		To 2
	Including 1	Not Including 1	Not Including 2	Including 2
Inequality:	x ≥ 1 "greater than or equal to"	x > 1 "greater than"	x < 2 "less than"	x ≤ 2 "less than or equal to"
Number line:
Interval notation:	[1	(1	2)	2]

Example: to include 1, and not include 2:

Inequality:

x ≥ 1 and x < 2

or together: 1 ≤ x < 2

Number line:

Interval notation:

[1, 2)

More Examples

Example 1: "The Nothing Over $10 Sale"

That means up to and including $10.

And it is fair to say all prices are more than $0.00.

As an inequality we show this as:

Price ≤ 10 and Price > 0

In fact we could combine that into:

0 < Price ≤ 10

On the number line it looks like this:

And using interval notation it is simply:

(0, 10]

Example 2: "Competitors must be between 14 and 18"

So 14 is included, and "being 18" goes all the way up to (but not including) 19.

As an inequality it looks like this:

14 ≤ Age < 19

On the number line it looks like this:

And using interval notation it is simply:

[14, 19)

Isn't it funny how we measure age quite differently from anything else? We stay 18 right up until the moment we are fully 19. We don't round up to 19 from 18½ onwards.

Open or Closed

The terms "Open" and "Closed" are sometimes used when the end value is included or not:

(a, b)	a < x < b	an open interval
[a, b)	a ≤ x < b	closed on left, open on right
(a, b]	a < x ≤ b	open on left, closed on right
[a, b]	a ≤ x ≤ b	a closed interval

These are intervals of finite length. We also have intervals of infinite length.

To Infinity (but not beyond!)

We often use Infinity in interval notation.

Infinity is not a real number, in this case it just means "continuing on ..."

Example: x greater than, or equal to, 3:

[3, +∞)

Note that we use the round bracket with infinity, because we don't reach it!

There are 4 possible "infinite ends":

Interval	Inequality
(a, +∞)	x > a	"greater than a"
[a, +∞)	x ≥ a	"greater than or equal to a"
(-∞, a)	x < a	"less than a"
(-∞, a]	x ≤ a	"less than or equal to a"

We could even show no limits by using this notation: (-∞, +∞)

Two Intervals

We can have two (or more) intervals.

Example: x ≤ 2 or x >3

On the number line it looks like this:

And interval notation looks like this:

(-∞, 2] U (3, +∞)

We used a "U" to mean Union (the joining together of two sets).

Note: be careful with inequalities like that one.
Don't try to join it into one inequality:

2 ≥ x > 3 wrong!

that doesn't make sense (you can't be less than 2
and greater than 3 at the same time).

Union and Intersection

We just saw how to join two sets using "Union" (and the symbol ∪).

There is also "Intersection" which means "has to be in both". Think "where do they overlap?".

The Intersection symbol is an upside down "U" like this: ∩

Example: (-∞, 6] ∩ (1, ∞)

The first interval goes up to (and including) 6

The second interval goes from (but not including) 1 onwards.

The Intersection (or overlap) of those two sets goes from 1 to 6 (not including 1, including 6):

(1, 6]

Conclusion

An Interval is all the numbers between two given numbers.
It is important to show if the beginning and end number are included.
There are three main ways to show intervals: Inequalities, The Number Line and Interval Notation.

451, 454, 1073, 1074, 1075, 1076, 2316, 2317, 2318, 2319

Footnote: Geometry, Algebra and Sets

You may not have noticed this ... but we have actually been using:

Algebra,
Geometry (the Number Line is a line), and
Sets (an interval is a set of numbers)

all in one subject. Isn't mathematics amazing?