Here we will look at the real number line.
This is kind of the set up. We have this line with numbers on it.
The point zero
is known as the origin.
The distance between any two points
is called the scale.
The next thing that deals with the number line is inequalitites.
The way these work
If a number a
is to the left of b on the number line
We have two numbers. The number on the left
We say a is less than b
So, for example
3 is less than 7, since 3 is to the left of 7
We also have that 8 is less than 11
If a is to the right
of b
then we say a is greater than b
Since 9 is to the right of 4
9 is greater than 4.
Also, since 3 is to the right of 1
3 is greater than 1.
If they are at the same location
so if they are at the same spot on the number line
In this case, we say a is equal to b.
For example, 5 is equal to 5.
We also have a little more.
If the number a
is either less than or equal to b
then we can write this as
a is less than or equal to b. For example,
3 is less than or equal to 7.
We also have 5 is less than or equal to 5.
Similarly,
We can also have a is greater than or equal to b.
This happens if a is either greater than or equal to b.
For example,
9 is greater than or equal to 4
5 is greater than or equal to 5.
Since 5 is equal to 5, and that is included.
All of these symbols together are known as inequality symbols.
The less than, greater than, less than or equal to, and greater than or equal to are all inequality symbols.
In addition,
a less than b
means the same thing
as b is greater than a.
For example,
5 is greater than 2
or 2 is less than 5
We can switch the order, as long as we switch the inequality symbol.
The next thing we will look at is graphing these inequalities on the number line.
There are going to be two main rules.
We are going to use an open or a close parenthesis
for less than or greater than
These are known as strict inequalities.
because they have to be strictly greater than
or strictly less than
We don't include the equal to case.
The other rule
says we will use square brackets
for less than or equal to
or greater than or equal to
And these are non-strict
inequalities
So, let's do some examples.
We have that 5 is less than x.
We start by looking at the number, which is 5.
We see where that is on our number line.
The fact that this is a strict inequality tells us we will use a parenthesis.
We say that 5 is less than x.
Which can also be written as
x is greater than 5.
So, we want the numbers that are bigger than 5.
We will put an open parenthesis at 5.
And we will shade all the numbers that are bigger.
This is the graph of the inequality.
Another example, 7 is greater than y.
We can also write this as y is less than 7.
We want numbers smaller than 7.
Since this is a strict inequality,
We will put a parenthesis at 7
And we will shade all the numbers below it.
So there is the graph of this one.
2 less than or equal to a
can also be written as a is greater than or equal to 2
So, we want the numbers greater than or equal to 2.
Since this is a non-strict inequality, we use a square bracket.
And we want the numbers bigger than, so we will shade to the right.
One last example,
We will look at b less than or equal to 4.
This is a non-strict inequality
So we will use a square bracket
And this says we want the numbers less than 4.
So we will shade to the left.
So, there is the graph of this one.
