INEQUALITIES AND THE NUMBER LINE

• Inequalities
• Opened and closed sets
• Sets on the number line
• Go to the number line/negative numbers home page
• contact me

In the above problem, if the potential price of the gift about to be bought was designated as our solution, then the solution is not just one number, but entire range of numbers (at least it is before you go shopping).
To illustrate this further, let the letter p represent "the potential price of the gift about to be bought."
Then p < 100 or p = 100

We'll be going back to this problem in just a minute, but first I have to introduce two new symbols: £ and ³.

£ is similar to "<" but it means "less than or equal to"
So if I wrote "n £ 6" then n could be 6, or any other number less than 6.
Possible values for n would include {6, 5, 3.2, 1/2, 0, -3} just to name a few.
However if I wrote "n < 6" then n could not be 6 itself, but only numbers less than 6.



³ is similar to ">" but it means "greater than or equal to"
So if I wrote "n ³ 6" then n could be 6, or any other number greater than 6.
Possible values for n would include {6, 7, 8.2, 9 1/2, 100, 3,000,000} just to name a few.
However if I wrote "n > 6" then n could not be 6 itself, but only numbers greater than 6.

Going back to our shopping example we could write:
p £ 100. [Remember, p represents the prices that we could afford for a birthday gift, where our maximum is $100]

Now this is well and good, but there are a lot of numbers less than $100.
$0.25 is less than $100 too, but unless your going to wrap up a gumball, you need a lower limit as well as an upper limit.

All right let's say you intend to spend at least $65.
Then we could say p ³ 65 (p could be 65 or any number higher than 65).
p ³ 65 is the same statement as 65 £ p (65 is less than p or 65 = p)
So we now have two statements regarding p: one a lower limit, and one an upper limit.
65 £ p and p £ 100
We could put these two statements together by writing 65 £ p £ 100.

All of the statements that you have seen involving <, >, £, and ³ are called inequalities Let's look at a few more examples of inequalities

Example 1
5 ³ n > -1

this statement has two parts 5 ³ n and n > -1

Let's look at the first part: 5 ³ n (which means the same as n £ 5)
This basically says that 5 = n or 5 is greater than n
Thus n could be 5, 4, 3, 0, -3, -4, etc.

The second part says n > -1 (which is same as -1 < n)
So n cannot equal -1, but it is greater than -1
Thus n could be -0.9, 0, 1, 2, 4, 5, 7, etc.

Putting the two statements together we have the range of numbers between -1 (not included) and 5 (included).

Example 2
-5 < n £ -1

-5 < n means that -5 is less than n, but n cannot equal -5.
Thus n could be -4.9, -4, -3, 0, 1, 2, etc.

n £ -1 means n is less than -1 or n is equal to -1.
Thus n could be -1, -2, -3, etc.

Putting both statements together we have the range of numbers between -5 (not included) and -1 (included)

We will be using the parenthesis and bracket symbols“(“, “)”, “[“, and “]”.
We’ll use “(“ or “)” when a number is not included in the range and “[“ or “]” when a number is included in the range.

Using this equipment let’s look at the inequalities mentioned in the preceding section

First we had p £ 100
This could be expressed as (-¥,100]

Notice that -¥ has a “(“ before it-We will always use the parenthesis symbols whenever we refer to -¥ or ¥.
Notice that the 100 has a “]” next to it since it is being included in our range.

Next we had p ³ 65
This could be expressed as [65,¥)

Notice that ¥ has a “)“ after it and the 65 has a “[” before it since it is being included in our range.

Putting these two lines together we had the inequality 65 £ p £ 100
which would be written as [65,100]

Note: if we use set notation we could write (-¥,100] Ç [65,¥) = [65,100]

In example 2 we had the inequality -5 < n £ -1
Which would be written as (-5,-1]

A number range enclosed in parenthesis is called an open set.
One example of an open set: (1,3) meaning all of the numbers between 1 and 3 where 1 and 3 are not included, and equivalent to the inequality 1 < n < 3.

A number range enclosed in square brackets is called a closed set.
One example of a closed set: [1,3] meaning all of the numbers between 1 and 3 where 1 and 3 are included, and equivalent to the inequality 1 £ n £ 3.

Okay we have all of our equipment, it's time to dive into the pool.

Graph n ³ 2

|___|___|___|___|___|___|___·___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5® 

|___|___|___|___|___|___|___o___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___o___|___|___|___·___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___·___|___·___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___·___|___o___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___o___|___·___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___o___|___o___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___·___|___|___·___|___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___o___|___|___·___|___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___·___|___|___o___|___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___o___|___|___o___|___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___·___|___o___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___o___|___·___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___o___|___o___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___·___|___·___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___|___|___|___|___·___|___·

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___|___|___|___|___o___|___·

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___|___|___|___|___·___|___o

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___|___|___|___|___o___|___o

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___o___·___|___|___|___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___·___o___|___|___|___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___o___o___|___|___|___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___·___·___|___|___|___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

·___|___|___|___|___|___|___|___|___|___·

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

o___|___|___|___|___|___|___|___|___|___·

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___|___|___|___|___|___|___·

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___|___|___|___|___|___|___o

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

o___|___|___|___|___|___|___|___·___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___|___|___|___|___·___o___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___o___|___|___|___|___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___|___|___|___|___|___o___o

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

·___|___|___|___|___|___|___|___|___|___·

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ® 

|___|___|___|___|___·___|___|___|___|___|

¬ -5  -4  -3  -2  -1   0   1   2   3   4   5 ®