Let's talk about interval notation.
Here's an example of an interval from 3 to 5, including the endpoints.
The first thing I'd like to do is draw a graph on a number line.
Here are my endpoints: 3 and 5.
This interval includes all the real numbers x
between 3 and 5 and also includes the endpoints.
I'm going to shade all those numbers between 3 and 5.
Because the endpoints are included, I want to use square brackets.
Now let's think about the inequality.
The real number x in this interval
has to be less than 5 or equal to 5.
It also has to be 3 or higher,
so it's greater than or equal to 3.
For the second example, we have the inequality given.
I like to go ahead and look at the graph
on the number line again here. There are my endpoints.
This tells me that x is a number between negative 2 and 1.
This time, I'm not going to include the endpoints.
We shade the numbers, and we use parentheses
to show that the endpoints are not included.
Now I want to write the interval. When you look at the graph,
you can sort of see the intervals staring right at you.
It's all the numbers from negative 2 to 1, not including those endpoints.
Notice again I used parentheses because the endpoints are not included.
Our third example has the symbol infinity in it.
The symbol is used to indicate unboundedness in the positive direction.
Again, I want to look at the graph.
I only have one number for an endpoint: negative 3.
That infinity again tells me that
I want an interval that's unbounded in the positive direction.
Any number that's bigger than negative 3 should be included, so I want to shade all of this.
Now what about negative 3? The parentheses tell me it's not included,
so I'm going to use parentheses over here as well because it's not included.
Now for the inequality, x is any real number
that is bigger than negative 3, so x is greater than negative 3.
Again, the endpoint is not included.
Here again, we have x is less than 2.
Let's look at the graph.
Where on the number line will we find numbers that are less than 2?
They are to the left.
Anything smaller than 2 will be on this side.
Again, it was not included, so I'm going to use parentheses.
Now what would the interval look like?
Since it's going to the left without bound,
this is unbounded in the negative direction, which is negative infinity.
It goes to the endpoint 2, which is not included.
Lastly, this interval is actually all real numbers.
It goes from negative infinity to positive infinity.
If we look at the graph on the number line, that's everything.
How would we write an inequality?
We actually could just write in words: all real numbers x.
The inequality would be negative infinity is less than x,
which is less than infinity.
Given an inequality, you should be able
to write the interval or draw the graph.
Given the graph, you should be able to come up with the inequality and the interval.
Given the interval, you should be able to get the other two.
This is an important skill that you need to practice until you understand it.
Now let's look at solving inequalities.
Here are two inequalities to solve.
The first one is a linear inequality.
The goal is to isolate the x
on one side of the inequality symbol.
My first step: subtract 6 from both sides,
so 7 minus 6 will leave me with 1.
Now I want to distribute the 5.
Subtract 5 from both sides.
1 minus 5 will give us negative 4.
Here's where you want to be careful.
You're going to divide both sides by a negative number,
so remember to flip the inequality symbol: reverse it.
That's going to leave me with x is
greater than or equal to positive 4/5.
This is in inequality form, and I want my answer to be in interval notation.
This is all real numbers x that are greater than or equal to 4/5.
4/5 is my endpoint.
I want anything bigger than 4/5, so that's towards positive infinity.
I want to include the 4/5, so I use a square bracket.
Now our second example is actually a compound inequality.
Notice it has two inequality symbols.
The goal here is to isolate the x in the middle.
Remember that whatever you do to the middle
you also need to do on the left and on the right.
My first step: multiply everything by positive 5.
Negative 1 times 5 is negative 5.
I'm going to have the 2x minus 4 in the middle.
5 times 0 is of course still 0.
Add 4 to all the parts of the inequality.
That's going to give me negative 5 plus 4, which is negative 1.
There's 2x. If I add 4 to 0, I get 4.
Now I'm going to divide by positive 2.
Inequality symbols stay the same.
4 divided by 2 is 2.
Again, I want my answer to be in interval notation.
Notice that the solutions to the inequality
are all the numbers x between negative 1/2 and 2, not including the endpoints.
Use parentheses because the endpoints are not included, and there's your interval.
Be sure to practice solving these inequalities
and using interval notation until you're comfortable with the process.