This is video number four for sections
3.5 and 11.1 --
solving absolute value inequalities.
OK so now we're going to put
everything together that we've done --
we're mixing in the
absolute value bars along with the
inequality symbols.  How would I
solve something like this?   And the good
news is
it's basically the same idea that we
used in those
absolute value equations.  If you can
isolate the absolute value
and then if you can think in terms of
value,
you can solve the problem.  Let me, let me
show you how this is going to work.
In this particular problem the absolute
value bars
are already by themselves -- that's great,
there's no other stuff
on the outside, no other junk that I
need to get rid of.  So
I'm able to immediately start thinking
in terms of
mystery number -- I have a complicated
mystery number, it's two x plus one.
So there's my mystery number.  I don't
know what that mystery number is but
let's think about what i do know.
The value part of that mystery number
is less than or equal to seven.  Now
that's a little different than we had
before.
If it only said that the value part was
equal to seven --
well I know there's only two spots you
can go where the value part
is equal to seven -- either negative seven
or positive seven.
Another way to say it is -- because you can
only walk seven steps
to the right here or to the left, and end
up here.
The less than though says -- well wait a
minute you don't have to actually be
exactly seven steps away from zero.
You can be anything less than seven
steps.
So like right here for example, or right
here,
or right here -- really anywhere in this
section
you would have walked less than seven
steps away from zero.  Rr you or or you
could say it --
all of these numbers have value parts
that are below seven.
Now forget it out here, you know, once you
get to eight or nine,
those numbers, or negative 8 or negative
9, those numbers have value parts that
are
too big -- but anywhere in here: 1 2 3 4 5,
negative 1, negative 2 -- all of those value
parts are less than 7.
I think drawing it is so helpful because
now I get to the part where I list my
possibilities.  Well that listing is going
to look a little bit different here
because look at this picture -- it's not
just two dots
like we had before -- it's a whole section
of numbers, right -- this green shading,
that's where I know my mystery number is
located,
somewhere in that green shading.  Well
notice that this is a between picture --
my mystery number is in between negative
seven
and positive seven.  That's exactly what I
need to write over here --
my mystery number is between
(remember two less thans, one on either
side) --
negative seven, the smaller number, and
positive seven,
the bigger number.  And I think I need
equals, right -- because we've already said
it's OK, you can be
right at seven or negative seven, that
would still work because I've got an
equals
in the problem.  This is the listing of
the possibilities -- now I know where my
mystery number is,
it's between those two numbers.  Well this
is one of those compound inequalities.
This is the one where
as long as I do the same thing to all
three
sides,  I can solve the problem.  So
I'll start by subtracting one in my
effort to
get that x by itself,
and then it looks like I'll need to
divide everybody by two.
And what I'm going to end up with in the
end
is this statement, which is probably best
read as x is
between negative 4 and 3,
including negative 4 and 3 because of
the equals, but
x could be any number between negative 4
and 3.
What are we saying?  We're saying back in
that original problem,
if you picked any x between negative 4
and 3
you will make a true statement.
Just like before they may ask you to
draw a number line picture
of your answer -- that would be pretty
straightforward.
Remember this number line here was just
a tool, this is not part of the answer.
This number line would actually contain
the answer.  I would need negative 4
and I would need 3 and I'm shading in
all the numbers in between, including
the ends.  So here would be a number line
picture of the solution.
And the interval notation -- left number
negative 4
comma right number 3, and both in
brackets because I'm including
the ends.  That's solving
an absolute value inequality -- definitely
tricky, I mean
make no mistake.  And I think the tricky
part is because you have to do the
thinking, you have to use this
tool and think about the mystery number.
Let me show you one other example here,
so take a look at this example.
It looks very similar.  This time the
absolute value bar
part of the inequality is not by  --
it's got this
2 out here on the outside.  So I'm going
to have to divide both sides by 2
as a starting point.  You've got to get
the vertical bar part
by itself before you do anything else.
Once you do that, everybody in there is
trapped
in the vertical bar so don't touch it,
you've got to rid yourself next
of the vertical bar.  Here's my mystery
number -- it's x minus 3.
And let's see, what do I know about this
mystery number?
I don't know anything directly about it
but I do know that the
value part of the number is more
than two.  So if I draw my typical tool
here --
let's see, if it was exactly two then of
course I'd go out two spaces to the
right and I'd go out
two spaces to the left.  But I'm looking
for numbers
that have a value part bigger than
2.  Well these numbers would work: 3 4 5
6 7 -- they all have value parts that's
bigger,
that are bigger than 2.  Where else?  How
about numbers like
negative 3, negative 4, negative 5,
negative 6 --
the value part of those numbers
would be bigger than 2.  Or you can just
think in terms of counting spaces or
taking steps.
I just want to walk more than two steps.
Well if I walk more than two steps to
the right I end up over here, if I walk
more than two steps to the left
I end up over here.  Now it's time for me
to write down my possibilities -- now pay
attention to this.
In the last problem, after I used my tool,
I was sitting
and looking at a between picture.
That is not the case this time -- this is
not a between picture.
In fact these are two separate sections.
My mystery number -- I cannot write a
between statement --
my mystery number could be in this green
section,
that green section is all numbers that
are below negative 2.
Or my mystery number could be in this
green section, that would be all numbers
that are bigger than positive 2.
I used the tool in order to help me write
out
the possibilities.  Once you've made it to
this step
you're home free.  You just have to
isolate the x, that's going to be very
easy --
just add 3 to both sides in both of
these things,
and it looks like I'm getting a final
answer of
x is less than 1
or x is bigger than 5.
That's the answer to the problem and you
may be looking at that saying -- well
I don't want to write it like that, I
want to write it all nice and short
like we did in the last problem.  Well we
can't because this
wasn't a between situation, this is not a
between answer --
this is two separate groupings of
numbers:
x is less than one or x is bigger than
five.  If i was to draw a number line
picture
of the answer, I would have to go to one
and I would have to go to five,
and then I'm supposed to color in
numbers that are lower than one -- well
that would be all of these numbers over
here --
and numbers that are bigger than 5.
Notice that I'm using
open circles because I don't have equals
signs here.
I don't have equals signs here or here
because I didn't have one in the
original
problem, right -- two and negative two back
on my tool were
not included.  I had to be only
more than two steps away, so I'm using
open circles.
Here would be shading in for numbers
that are bigger than
5. This is not a between picture, this is
not a between answer, so I can't write it
that way.
The third way is maybe the most
challenging here -- for the interval
notation I'm going to have to write this
in two pieces because I have two pieces
to my picture.
So first piece runs from negative
infinity
down to one -- no brackets anywhere in this
problem because
I don't have any closed circles.  Second
section
runs from positive 5 down to
positive infinity.
And then when you have a situation like
this where you're using interval
notation with two separate pieces,
there's a math
symbol that goes in between.  It looks
like the letter
u, it's called a union symbol,
but it is a special symbol -- it's not
actually the letter u, it just kind of
looks like that.
In mymathlab when you get to these
problems,
look in the toolbar at the bottom and
you will see
a union symbol that you can type in
for these particular problems.
Solving absolute value inequalities --
little bit tricky, but
follow the steps.  Here's the good news --
there really are
only two types of these problems and
I've worked one of each for you.
The ones that start with a less than
are going to be "between" problems and
work like this first example.
The ones that start with a greater than
in the problem are going to be two
section problems
and work like the second example.  So
those are really the only two things
that can happen to you
and in that sense it makes the learning
of it a little bit
nicer.  Obviously you'll want to practice
this
so that you can become skilled at it.
