Graphing a linear inequality on the
number line. In this topic you're gonna
be seeing statements like x > 4
and you're asked to graph this on the
number line.
I want to kind of pick apart what this
is saying and then we'll do a few more
examples. So first things first this is
an inequality because it has an
inequality symbol so this guy right here
is known as an inequality symbol. An
equality symbol is just the equal sign
an inequality symbol it means something
is greater than or something is less
than another thing. So there are four of
them that you need to know there is the
greater than sign >
there's the greater than or equal to
sign ≥,  so if there's an underline
underneath it means greater than or
equal to. And then the less than sign <
and less than or equal to ≤
so when you're asked to graph the
inequality X > 4 the X
part is really just a placeholder
it's a variable and it could change it's
saying just highlight all the numbers
greater than four to show me on the
number line which numbers are greater
than four. I could honestly erase this X
and put in the words highlight the
numbers so highlight the numbers greater
than four. So if I make a number line
really quickly, I'm gonna go ahead and
put four right here just in the middle
because it doesn't matter, the numbers
that are greater than four are going to
be the numbers to the right of four.
The way that the number line works is it
starts at the negative side so it starts
negative infinity and it counts its way
up to positive infinity so if I drew
more numbers on this number line 3
would go here 2 would go here 1
would go here and the number line is
infinite so it goes on forever but we
only draw small portions of it at a time.
So here's my number line numbers that
are greater than 4 are gonna be these
guys over here
all the way out to infinity. Now we can't
draw to infinity that's not something we
can do but we can draw an arrow to show
that we mean that this goes to infinity.
So if I highlight the numbers that are
greater than 4 there we go
the only thing that's left to be
concerned about is what about at 4
itself what happens right here? Here is
where the difference of these two come
in. So greater than or equal to means
that you're including 4 greater than
with no underlined means that we're not
including 4 we only want numbers that
are strictly greater than 4 so 4.1 works, 4.000001
would work but 4
itself wouldn't, whereas this guy right
here four is included. The way that we
deal with this we use an open circle if
there is no underline and we'll use a closed circle if
there is an underline and it'll be the
same thing on the less than or less than
or equal to. So all we have to do here
we're not including 4 so we're gonna
have an open circle here and there it is
that's X > 4 graphed on
the number line. Let's go ahead and do a
few more examples all right so you're
gonna notice that the variable can
change in this problem, it doesn't matter
what the variable is, it's saying the
same thing: highlight the numbers. So on
this first one highlight the numbers
greater than 1.
There's no underline here so I'm gonna
use an open circle at 1 so here's 1
there's my open circle, greater than is
gonna be to the right of that number so
these are all the numbers greater than
one and that's it I've graphed that on
the number line. For the next one a < 10
I want to go to 10 and I want to plot an open circle because there's no
underline here at ten
and then we want the numbers that are less than ten.
So the way that the inequality
symbol works it points to the
to the smaller number so the big end is
at the big number and the small end is at the
small number so you know that ten is the
cap everything else is less than it
because the big ends are at the ten side
and if that doesn't make sense to you
another way of looking at it is the
alligator way so the alligator is like
an alligator mouth right here it the
small alligator eats the bigger
alligator some people like to look at it
like that, but I like to look at it as
the big end goes towards the bigger
number either way doesn't matter this is
less than so numbers that are less than
10 are all of these guys so I'm just
gonna come in with my pen and just
scribbly scribbly scribbly on Alex of
course you have a tool that does this
you don't have to use scribble use it
will highlight the whole thing for you
but on paper you do have to do some
scribbles. For X ≤ 2
I'm going to 2
and I am plotting a circle with that is
filled in because there's this
underline here, so we want to highlight
all the numbers that are less than or
equal to 2 so here's 2 right here,
closed circle. All those numbers are less
than 2. For this last example down here
we have b ≥ 3, so we're gonna go to 3.
I'm drawing a circle. Now there is an
underline here so I'm going to fill in
the circle and then we want the numbers
that are greater than 3 so that
would be the those numbers, those are the
numbers of greater than 3.
One last thing I want to mention is negative
numbers so people tend to kind of make
mistakes when they start thinking about
negative numbers on the number line so
let me just kind of do a few, like two
examples, where we talk about that. So if
I said Z ≤ -5 you're still doing that
same process of you go to -5
you decide whether it's an open or
closed circle based upon whether or not
there's an underline so there is an
underline which means it is a closed
circle so -5 closed circle
and then it's highlight all the numbers
less than -5. So regardless of
whether you are a positive or negative
number, less than is to the left and you
can remember it less than and left both
have that L sound, less than is to the
left, but you could also think of this as
debt if your 5 dollars in debt you are
in less debt than somebody who is 10
dollars in debt so you're shading all of
these numbers over here. If on the other
hand I said that, let's use C, if C is
greater than -7.
I come to -7 I'm drawing an
open circle because there's no underline
here and then numbers that are
greater than -7 are all the
numbers to the right of it so it would
shade in this direction. The nice thing
is, and this is kind of a nice little
trick, if you have the variable part on
the left and it has to be on the left so
if you have variable and then you have
the inequality symbol and then you have
the number over here it'll always point
in the direction that you need to shade.
So, if you have a greater than symbol
you're going to end up shading to the right see it points to the
right so you shade to the right if you
have a less than symbol then you're
shading to the left and it points to the
left so you know you're shading to the
left. If your variables on the other side
that's not going to work so if it were
number and then the box and then
variable then it would be opposite, but
as long as it's written in this format
here it is actually showing you the way
that you need to shade.
So that's all there is to this topic you totally got this!
