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Intervals

Interval: all the numbers between two given numbers.

Example: all the numbers between 1 and 6 is an interval

All The Numbers?

Yes. All the Real Numbers that lie between those 2 values.

Example: the interval 2 to 4 includes numbers such as:

2.1

2.1111

2.5

2.75

2.80001

⁷/₂

3.7937

And lots more!

Including the Numbers at Each End?

Ahh ... maybe yes, maybe no ... you need to say!

Example: "we allow boxes up to 20 kg in mass"

If your box is exactly 20 kg ... will that be allowed or not?

It isn't really clear.

I will show you how to be precise about this in each of three popular methods:

Inequalities
The Number Line
Interval Notation

Inequalities

With Inequalities you use:

> greater than
≥ greater than or equal to
< less than
≤ less than or equal to

Like this:

Example: x ≤ 20

Says: "x less than or equal to 20"

And means: up to and including 20

Interval Notation

In "Interval Notation" you just write the beginning and ending numbers of the interval, and use:

[ ] a square bracket if you want to include the end value, or
( ) a round bracket if you don't

Like this:

Interval Notation

Example: (5, 12]

Means from 5 to 12, do not include 5, but do include 12

Number Line

With the Number Line you draw a thick line to show the values you are including, and:

a filled-in circle if you want to include the end value, or
an open circle if you don't

Like this:

Example:

(0, 20)

means all the numbers between 0 and 20, do not include 0, but do include 20

All Three Methods Together

Here is a handy table showing you all 3 methods (the interval is 1 to 2):

	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 we would have:

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 would be more than $0.00.

As an inequality we would 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:

(0, 10]

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:

[14, 19)

And using interval notation it is simply:

[14, 19)

Isn't it funny how we measure age quite differently from anything else? We remain 18 right up until the moment we are fully 19. Whey don't we say we are 19 (to the nearest whole number) from 18 years and six months onwards?

Open or Closed

The terms "Open" and "Closed" are sometimes used for 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, +∞)

[3, infinity)

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"