This is video number one for sections
3.5
and 11.1 -- describing inequalities.
So we're approaching the end of our unit
on
linear functions and essentially what we
have in these last couple of lessons
is an answer to this question -- what
happens
if we redo some of what we've been doing,
but instead of using an equals sign,
we use one of these symbols?  These are
called inequality
symbols:  less than, greater than,
less than or equal to, greater than or
equal to.
What if we start using these symbols
instead of an equal sign?
So that's we're going to talk about in
this lesson and in the next one.
This video starts with just the idea
that these
inequalities can be described
in a variety of ways, and so let me show
you what I mean by that.
Here is a starting example, kind of a
funny looking graph --
it has a starting dot here, it goes up a
little bit,
and then it goes down forever, and the
arrow means of course that it goes down
forever.
If I were to ask you what the domain is
of this function (that's an idea we've
practiced before) --
remember domain is just a list of all of
the
x values that ever show up on this graph.
Well there are a lot of x values that
never show up on the graph like negative
1
and negative 2 -- even 0 is not on the
graph.
The graph doesn't start until this spot
right here, which would be an
x value of one.  Now after that
every possible x value after is
eventually used, especially because this
thing goes on
uh forever, right -- in a rightward and
downward direction.
So I might say something like -- the domain
is all numbers one and higher.
And that's fine, but how can we describe
that set of numbers mathematically?
Well it turns out there are three ways
you can describe
this set of numbers.  The first way
is using an inequality symbol.  That's one
of these four
symbols that we talked about up here.  And let's see, in this case I'm talking about
domain so I'll use the letter x.
And I would be saying, well x can be
anything
that is bigger than 1.
And actually it can also be equal to one
so I would have to put that on as well.
So
all x's that are greater than or equal
to one -- that's the inequality symbol
way of writing the domain.
A second way that you could write it
would be to draw a picture.   Now I'm not
talking about an
x y picture -- I'm talking about
a number line graph, so just a
simple number line extending in both
directions --
because remember I'm just here trying to
draw
a picture of the x's that ever go
on this graph.  Well let's see -- we said
one and above, so you would mark some
scale
like I've just done on your number line.
And then right on the line, not above or
below it but right on the line,
color in all the numbers that we're
trying to describe.  So
we're starting at one and then we're
saying everything
higher than one.  The
filled in circle here at the start tells
me -- okay I'm
starting at one, the arrow at the end
means well this goes on forever --
one and above.  So that's a number line
way
of describing this set of numbers.  The
third way
is called interval notation.  This one
might be
new to you.  I find that it's easiest to
get
interval notation from the number line
picture.
Here's how it goes.  Take a look at what
you drew on the number line picture -- I'm
talking about the red
graph.  Write down first
the number that's farthest to the left.
Well that would be
one -- the number farthest to the left on
my red picture is a one.
Then put a comma and then put the number
that you colored in
farthest to the right.  Now that's kind of
a strange
question because this arrow means it
goes on forever.
Well we have a name for that in math --
it's called infinity.
Infinity is like a sideways eight, that's
the symbol for infinity.
This means oh -- all numbers from
one to infinity.  Then
around the numbers you either put a set
of parentheses
or a bracket.  Most of the time you use
parentheses; you only use a bracket
if you have a filled in circle,
so infinity does not have a filled in
circle -- infinity always gets a
parentheses.
But that one has a filled in
circle and so I put a bracket around it.
The bracket
communicates "including one."
OK that's the meaning of the bracket,
that one is included
in your set of numbers.  If I put a
parentheses there
it would mean -- no not one, only the things
more than one.
This is called interval notation; it
means the same thing as all these other
things -- it just means
all the numbers starting at 1 going all
the way
toward infinity.
Three ways of describing the same set of
numbers.
Let's try a second example.
I didn't make a graph or anything this
time, but let's say that we were looking
at some
function and we figured out that
the domain was
all numbers between negative three and
five.
And when I say not including the ends, I
just mean
not negative 3 and not 5.  So let's try
to describe
that set of numbers in those same three
ways.
First with the inequality symbols --
between, you may remember, has its own
special notation.  And the way between
goes
is you take your variable  (I'll assume
I'm still dealing with x here)
and you put it between two
less than symbols -- it's always an x or
whatever variable
between two less than symbols.  Then the
smaller number
goes on the left -- that would be the
negative three, and the bigger number
goes on the right.
Let's see do I need the little equals
signs?  I think I
don't because I'm not including
those ends.  So here would be the final
statement, and you can even read this -- I
mean some people try to read it
negative 3 is less than x is less than 5 --
that's true, but it's not very helpful.
The helpful way to read it is
x is between negative 3
and 5.  So that's how you would write it
with inequality symbols.  What about
a number line graph?  Well I'm going to
have to come over and make some scale on here.
I definitely need to go all the way to
negative 3 and I need to go to at least
five over here, so I've
drawn a number line,  I've filled in
sufficient scale.
Let's see, and I'm trying to color in all
the numbers
between negative 3 and 5.  I don't think
I'm going to have an
arrow on this one -- I'm just going to be
coloring in
in between those numbers and then
stopping.  Now here's the thing --
we're not including the ends this time,
so
on the picture, instead of doing a filled
in
circle, you do an open circle -- an
open circle deals with the fact that I'm
not including the negative 3.  I'm still
going to color in
everything in between -- you know all this
stuff is part of the answer -- remember
it's
right on the line, it's not above it,
you're shading the actual line.
And of course I'm going to go all the
way down to the five,
and what am I going to put at the 5?  Well
I'm not including the 5 either so I
would need an
open circle there at the 5 as well,
and there is my final graph.
Again the open circle is because I'm
not including the 5.  So here's a picture
of all numbers between
negative 3 and 5.  Lastly
let's see if we can figure out the
interval notation for this set of
numbers.
Remember the easiest way to get the
interval notation
is to look at the number line picture.
I take the number that is
part of that red graph farthest to the
left, that would be the negative three,
then a comma, the number that's farthest
to the right,
that would be the five.  And then
surrounding them I either put a
parentheses or a bracket -- remember the
only time you use a bracket
is if you want that end to be included.  I
don't
want those ends to be included (and sorry
this should say negative 3) -- I don't want
those ends to be included
so I would put a parentheses here and
I would put a parentheses here.  This
would be the interval notation.  I will
tell you one very
unfortunate thing about this notation is
this kind of looks like an ordered pair,
right -- it is not an ordered pair, it has
nothing to do with an
ordered pair.  In the world of
inequalities
this means all numbers between negative
3 and 5 not including the ends.
We'll continue to practice with these
three ways of writing
inequalities as we go through the lesson.
