This is video number three for sections
3.5
and 11.1 -- solving absolute value
equations.
This is another extension
of what we've been doing with linear
functions.
You've probably seen these vertical bars
before.  They're called
absolute value.  And in math
we use these vertical bars when we want
to talk about,
not the entire number, but
just the value part of the number, in
other words
if we want to ignore the sign.
So I don't care whether it's positive or
negative, I just want to know --
what's the value portion of the number?
The official definition of absolute
value I've got written here --
it stands for, technically, how many steps
you would have to walk from 0 to that
number
on a number line.  It doesn't matter
whether you're moving to the left
or to the right, so you can ignore left
and right --
you're just counting spaces.  So as a
couple of
very simple examples, this is read
the absolute value of negative four.  And
like I was saying, I just think of it as --
well if I strip away the sign, what's
left?
Well if you take off the negative then
the only thing left is the four.  The
absolute value of negative four
is four, that's the value portion
of the number.  If you're a visual person
you could do it the counting steps way.
Here's zero and here's negative four --
you're basically just being asked -- if I
was walking from zero to negative four,
how many spaces would I have to go?  I'd
have to go four spaces.
How about absolute value of positive
three.  Well if you take off the positive
all you're left with is three -- it's the
value portion of the number.  Again you
could count that out
on a number line.  In order to walk from
zero to three
it would take you three steps: one, two,
three.  The only kind of strange one
is zero -- zero doesn't really have a sign
to strip off,
but I mean if I started at zero and I
said  --well how many steps is it going to
take me to walk to zero?  It wouldn't take
me any steps, it would take me no steps.
I think a lot of people remember this as --
oh everything turns positive
except in the case of zero, and that's
okay but I would encourage you to really
focus on this --
that what you're giving is the value
portion of the number.
Watch how that helps in these problems.
Because the goal in this video
is to solve an equation
that has one of those absolute value
statements in it.  And the question is -- how
do you deal with that?
To be honest it's a little tricky
because it's
hard to figure out how to get rid of
those vertical bars.
I've given you kind of the main steps
over here --
you always start by just getting the
absolute value
part of the equation by itself, okay -- get
rid of all the other outside stuff.
And then there's not really an algebra
step
for getting rid of the bars, and I think
that's what bothers people about this --
it's really more of a thinking
step and using that idea of value, so
I will show you.  OK take a look at this
example problem.
What's between the three and the bar?
It's an
understood multiply, you know kind of
like when you have a 3 next to a
parentheses or something like that.
So I want to get that vertical bar
part by itself -- that's going to take me a
couple of steps,
I'll have to add 6 to both sides,
and then I will have to divide out both
sides by that
three.  And so the vertical bar part is
now sitting there alone,
that was step one -- isolate that absolute
value.
OK, now that I've done that, here's the
way to continue, here's the way to think
about what happens next -- it's a thinking
step
more than an algebra step.  Whatever's in
the bar
is trapped in there at the moment --  I want you to think of it as some
mystery number.  I've got this
mystery number and I don't know what the
mystery number is,
but I do know that the
value part of the mystery number is 5 --
that's what this equation says: the value
part of that mystery number
is 5.  So then off on the side I can say --
well let's see,
where could I find numbers whose value
part is five?  Well look there's only two
places you can go -- you can either walk
five spaces to the right
or you can walk five spaces to the left.
Those are the only two numbers
that have a value portion of five.
OK well that's what I mean by list
possibilities.
My mystery number therefore must either
be negative 5
or positive 5.  Well I'm done --
that's the solution to the equation.  I
should say solutions,
right -- there are actually two different
numbers that you could plug
back into the original equation in order
to make it true.
You get rid of the bars by doing this
thinking
step and listing possibilities.
Let me show you one that's just a little
bit more uh involved.
So here's a second example -- same task,
solve for x, it's got the absolute value
bars.
First step is probably obvious -- I need to
get those bars
uh by themselves.  So there's only one
reason it's not by itself right now and
that's that two,
so I'm going to divide both sides by two.
So that'll give me absolute value 3x
plus 1
equals 4.  And then, you know, I know
when students are first learning this,
the tempting next step
is to say --  oh I better subtract 1 from
both sides.  Well
you can't do that because this 1, this 3,
they're all
trapped inside the vertical bars, and
until you get rid of those
you really can't go any further.  How are
you going to get rid of them?
By thinking in terms of mystery number
and value and then listing the
possibilities.
Notice my mystery number is a little bit
more complicated this time --
it's the entire 3x plus 1.
I have no idea what that mystery number
is,
but I do know that the value part
of that mystery number is 4.  So
I go to a number line, I think -- well where
could I be?   There's two choices -- either
I'm on the left side
or I'm on the right side -- those are the
only two numbers
that have a value portion of 4 --
negative 4 and positive 4.  Another way
to say it is --
if i want to walk four spaces, there's
only two places I can end up from zero:
at negative four, four spaces to the left,
or at positive four,
four spaces to the right.  I use
that in order to list my possibilities
for the mystery number.  So
my mystery number, 3x plus 1, could be
negative 4,
or my mystery number, 3x plus 1,
could be positive 4.
Mystery number and value -- once the bars
are gone,
now you can touch the one and the three
and all those normal algebra things.  In
fact you're going to have to do it twice
in order to get the solution.  This number
line that we drew right here is not part
of the answer, it's just a
tool we're using to help us identify
those possibilities for the mystery
number.
OK now we're almost done -- just do the
simple algebra steps --
subtract one from both sides in
both
cases here, and that's going to give me,
let's see, a 3x equals negative 5
here, and 3x equals 3 for this one, and of
course divide both sides by 3,
and I'll end up with my two values
of x.  You might say that
we essentially looked at this problem as
having two
mystery numbers.  I mean we wanted to
solve for
x but before we could ever do that we
had to essentially figure out what
3x plus 1 could, could be equal to,
and then we were able to take it down
and figure out the values of x.
Don't lose sight of what we're saying
here -- if you go back to the original
equation, we're just saying that either
of these two numbers,
if you were to plug them in and work out
the math,
either of these two numbers and only
these two numbers would make this
equation true.
