Okay.I wanna show you
a much better way,
an easier way, to represent
inequalities.
Let's start about talk talking about sets.
First A.
Set A intersected with set B
The intersection of set A and set B first. In this case
is defined
as hard as it may seem as all the members
x where the set of all x
such that, (that means "such that", that straight line) such that
x is
an element of A (that's what the "e" is)
and also
is an element of B. Well do you see that there's only two members that are both
an element of A and of B?
So the intersection of A and B is the set
containing 2 and 4. That's important for you to understand what intersection.
mMst people can find that.
Now we want to
differentiate intersection with something completely different and that
is a union.
A union B
is not going to be 2 and 4. Well
A union B is going to be defined as (it's the same definition except the and
changes to an or)
the set of all x's such that
X is an element of A (Remember what element is)
or it's an element of B.
So as long as you're in one or the other you get into the union
and this answer (please notice
as long as you're in one or the other you're in there.) {1,2,3,4,
6,8} and notice that two and four even though they were in both are not listed
twice.
Okay? We list everything once. So hopefully you'll see the difference
between
an intersection and a union. It's much easier to be in the union.
Now up to this point
we graphed
something like this set: all the X such that X is greater than or equal to
2
with a picture like that.
We put a a filled in circle on 2,
didn't we, and we went to the right because X was greater.
I would like you to consider
instead of using the filled in circle we're going to use
from now on (I'll show you why in a second) a bracket.
Okay. In this one
this would be X such that X is less than
or equal to 1 that would have been a filled in circle as well
so we're gonna use a bracket there.
Now in this one it's not a filled in circle.
X is less than three, okay? We could use
an empty circle. What I'm gonna have you do instead of
the empty circle or the bracket because it's not an or equal to
I'm gonna use a rounded
bracket. I'm probably should call it a parentheses, right?
That's what we're going to use. Okay.
So let me summarize when we have a greater than or equal to
or less than or equal to
that means we want to include the endpoint
the or equal to we're gonna use a bracket. Okay?
Make sure it goes the right way!
And when it doesn't include the or equal to like a less than
or a greater than and we don't want to include the endpoint we're going to use
rounded parentheses.
Now that 's going to buy us a whole lot of...
At least we're going to have WAY less effort I'll show you why.
Okay, let me show you what these brackets are gonna buy us. Normally when we
graphed or represented this
conjunction it look like that. All the numbers between -2 end 4 and of course
filled in dots because of the or equal to's Now I'm gonna ask you to draw it
differently.
With brackets, now,
that doesn't buy you much. You might think you still have to do that drawing
and the drawing takes a long time. Okay? I can represent this without going to the
trouble of a drawing using what's called
interval notation. Interval notation
is using the correct bracket or parentheses
and showing the leftmost point
and the rightmost point. Now in this case
they're both brackets. The leftmost point is -2
and the rightmost point is 4 so you can save yourself drawing this picture
by just using this notation.
[-2,4]. Let's look at another one.
This is... if we drew our old type of picture
notice that the one would have an
empty circle and the five would have a filled in circle, wouldn't it, because of
the lack of
an inquality, an equal sign around the one.
Well, if we start using our brackets
and parentheses
you could draw the picture like that but remember our goal here is to not have to
draw this big picture
and we're going to use interval notation
with a rounded bracket
next to the one and a square bracket
next to the five because of course there's no or equal to next to the one.
Okay? Now, this one's going to be kind of just the opposite
with different numbers. Okay? We don't have an equality sign next to the zero
so the zero gets
the circle that's not filled in
and the negative 4 gets a filled in circle cuz he has an or equal to.
Now, if we write this... Well, first lets graph it
with the interval notation.
Now let's do it the easy way with interval notation with the lowest
or left most point and the
highest or right most point. We get this answer.
Okay, once again note that the bracket is next to the negative 4 is the squared off
cuz he's an or equal to
and I have a rounded parentheses next to the 0 cuz he doesn't have an or equal to.
I do not want to include the end point there.
Hopefully you got the idea. Now some of these inequalities you may remember
go on for ever and if we were to look at a picture of this one
X is greater than or equal to 2
we're gonna have to involve infinity. The leftmost number is negative infinity
and the rightmost number is positive infinity.
This picture would look like this
so its left most point
is 2 but it's right most point
is infinity and we're thinking we never can really make it to infinity
so it's not going to be an equal sign. So anytime you use infinity you're going to use
a rounded
parentheses so the interval notation that represents this picture
is a squared-off bracket for 2
and a rounded bracket for positive infinity.
Remember with infinity we're round it since we never really make it there.
Now this one if you draw the picture
is gonna look like this. The leftmost point
is not positive infinity. It's negative infinity
and we never make it there so we're gonna have a rounded
on that one and we're also in this case going to have a rounded one on the zero
because it's
not an or equal to so that's going to save you drawing that whole picture so
there's your interval notation
representation of this picture.
Hmmm. Now what are we gonna do with this one? If we wanna write it in interval
notation
because it's a disjunction it's an or and we need a leftmost point and a
rightmost point and there's a hole in the middle.
Well there's a trick. That's why I'm showing it.
Well you can do
some representation of it but you can't give me
the exact leftmost and rightmost point where you
can give me those two which it but there's a hole in the middle.
we're going to use the Union concept.
This is the answer believe it or not. Now
where did that come from? I want you to see that the interval here
represents the leftmost picture there
from negative infinity up to -1 and the other part of it
is represented here from 6 to positive infinity
and since we can't really have a a whole picture without a hole in it
if you would we're gonna just use the Union and put the two
mini pictures together. Okay? So anytime you have a disjunction you're gonna have
to use
the union.
Okay? Well, let's solve one
and get an answer. To solve this basically it's two little problems
and we'll solve them separately.
Okay?
The left problem we'll subtract 5 or add a -5 from both sides
and then we'll divide both sides by three.
Now this is the big leagues;
sometimes things don't come out evenly 5/3
Where is 5/3 on the number line? Well, you should read it as a mixed
number
one and two thirds, right?
So it would kinda look like that,
wouldn't it? And this is a disjunction so it's an or
so I'll solve the other problem separately. Subtract 1 from both sides
and how do I graph X is greater than 3?
That's going to be a rounded
end point there
and I'm gonna have to use... I'm gonna have to...
I'm gonna need to use union man, okay?
and basically we just union the two answers. I hope you appreciate that.
From negative infinity to 1 2/3
union with from 3 to positive infinity.
Note also that every time we use infinity of any kind
we have a rounded parentheses.
Okay?
Now here's a conjunction, okay? A conjunction when you have to solve it
really has three sides.
So whatever I do
I'm gonna have to do three times.
To get x alone I have to subtract three so I'll do it to all three sides.
Okay? Now
I need to graph this. Basically it's all the numbers from -2
to 5 and it's inclusive because they're
or equal to. So this is pretty easy to write in
interval notation.
Just brackets
around both because it's or equal to. The lowest number is -2 or
leftmost and the rightmost number is five and there's your answer.
Let's do another conjunction solving,
solving it as a three-sided equation.
We add five to all three sides.
Now how do I get rid of 1/2?
Well
I'll multiply by the LCD and I'll get
X is greater than 6
and x is less than 12 but there's no or equal to here so I use a rounded
bracket
and it's a conjunaction. Okay let me do a couple tricky ones.
I gotta get x alone.
I have fractions. Remember the trick for fractions:
multiply everybody by the LCD. So in this case
that 3 will nicely go away and I end up with
what I have here and I still have to get the x alone so I'll
add one to all three sides
and then divide by what?
Divide by -2 because that's who's multiplying the x.
But when I divide by a negative number
what do I have to do? It's time to flip the signs
when I divide by a negative number. There was no escaping it
and if I do that I get X is smaller than 4 and greater than -5. You have to
read this backwards.
X is smaller than four because
the mouth is opening to the 4 and x is greater than -5
cuz the mouth is opening to the x. This is going to be a conjunction.
The numbers smaller than four
and at the same time greater than -5. Well,
what's both pink and blue is the intersection isn't it? So that's a
picture of the answer.
And that's
the interval notation. Okay. Doesn't get much harder than that!
And let me change this problem just a tweens
and it's going to change everything. The same problem except the signs have been
changed to the other way.
Okay?
Still going use your LCD man;
still gonna add one to everybody;
still gonna divide by -2
and since I divide by -2
What am I gonna do? I'm gonna flip the signs.
Now you gotta be careful. What is
both because it's a conjunction
greater than 4
(X is greater than 4 the mouth is opening up to the x, isnt it?)
and also less than -5.
Well,
there is no intersection; there is nothing that is both blue and pink.
So, guess what the answer in this case is?
There is no... The answer is
the empty set. Sometimes that can be the answer so be looking for that.
Well, okay, go try practicing this.
It's kinda challenging. Anything can happen, okay? And
be watching for the difference between a conjunction and a disjunct.