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1370397671dect_rerichsen
>> Good morning or afternoon
or evening,
whenever you're working
on this section.
We only have one section
in Chapter 9
that we will be covering,
which is 9.3,
which covers equations
and inequalities involving
absolute value.
We are using the Blitzer
Introductory
and Intermediate Algebra text,
and I am Professor Shannon
Gracey from MiraCosta College.
So, here we go.
We'll be looking at how
to solve absolute value
equations,
how to solve absolute value
inequalities in two formats,
less than and greater
than format, and talking
about absolute value
and equalities
with no solution
or infinitely many solutions
and then solving applications
using these skills.
So, warming up,
in Introductory Algebra,
you would have covered,
[Pause] I think in Chapter 2.
I believe it's Section 2.7.
So if you need a little bit
of practice on inequalities,
if you go back
to the 2.7 video from Blitzer,
that should get you caught up.
And that might be a good idea.
A lot of times people have
some trouble
with that initially
and going back
over that will be beneficial.
So, let's warm
up by graphing the solution
of this inequality.
Go ahead and pause the movie
and do your thing.
[ Silence ]
Ok. So, here we go.
We want to graph all numbers
between negative 6 and 6.
That is the solution
for this inequality.
We do not include negative 6
or 6 because there's not a
line under the inequalities.
So the way you graph this,
is you put a parenthesis
around the end point
and then you shade the numbers
in between to show
that any number
between negative 6
and 6 will make this a
true statement.
Now, Interval Notation,
you know, they didn't ask
for this, so don't worry
if you didn't put it down,
but Interval Notation,
the solution would be
from negative 6 to 6.
So these are the answers
for this problem.
So that the graph is the
picture and the interval
notation says all values
between negative 6 and 6,
non-inclusive.
All right.
So here we go.
How would one rewrite an
absolute value equation
without absolute value bars?
So if C is a positive real
number and U represents any
algebraic expression,
[Pause] then [Pause] absolute
U equal to C is equivalent
to U equal to C or U equal
to the opposite of C
or negative C.
So if you take a look
at the absolute value
of X equal to 6, equal to 6.
If you recall,
absolute value represents the
distance from zero
on the number line.
So what this equation is
actually saying is
which number
or numbers are six units away
from zero on the number line.
So, here we go.
If you look, and if we were
to actually graph this,
but you don't usually graph
them in this form,
but do you see that 1, 2, 3,
4, 5, 6, you could actually
plot that number?
Right? That single value.
That's six units away.
And then 1, 2, 3, 4, 5, 6,
negative 6 is six units away.
So when you're solving these,
you would say, oh, well,
X would equal to 6
or X would equal
to negative 6.
So the U in this statement is
X and the C
in this statement is 6.
So in Roster Notation,
this one would go
in Roster Notation
because there's only two
values, so we can just list
the values.
And so, there we go again.
We have the picture
and we have the solution.
[ Silence ]
All right.
So next step [Pause] we have
something just a wee bit
more complicated.
[ Silence ]
The absolute value
of X minus 3, equal to 6.
So with absolute value
equations, you always want
to make sure you have the
absolute value alone
on one side
and everything else
on the other side.
So, we have
that situation here.
All right?
Now, let's take a look
at this one.
We still want
to be six units away
on the number line,
but now it's six units away
from this quantity, X minus 3.
So, if you think about it,
this quantity inside,
this is our U. X minus 3 has
to be equal to 6
or X minus 3 has to be equal
to negative 6.
So if you solve this,
it ends up with two very
simple linear equations
and one variables.
So if you add 3 to both sides
on this first equation,
you get X is equal to 9.
And, over here, if we add 3
to both sides,
we get X is equal
to negative 3.
[ Silence ]
So now, [Pause] it's always
kind of a good idea
to check [Pause]
your solutions.
So you check it
in the original equation.
So if we're checking X equal
to 9, we would have absolute 9
minus 3.
Is that equal to 6?
Well, absolute 6.
Is that equal to 6?
Yes, because the absolute
value of 6 has the result of 6
and 6 does, indeed,
equal to 6.
And then, similarly,
if you're checking [Pause] X
equal to negative 3,
[Pause] we would have [Pause]
absolute negative 3 minus 3.
Is that equal to 6?
Absolute negative 6.
Is that equal to 6?
6 equals 6 so we're done.
They both check out.
Once you know they both check
out, you can go ahead
and put your answer
in Roster Notation
and the two answers were
negative 3 and 9.
Now, if we wanted to look
at what a picture would look
like, we would have negative 3
here and 9 here.
So notice,
and I won't have you graph
those most of the time.
This was just sort
of an exploratory exercise
here, but I want you
to see the difference.
We found two distinct values
for X instead of an interval
of values for X.
And that's the difference
between absolute value
equations and inequalities.
All right.
So, here we go.
Example one has got four
examples here.
Why don't you go ahead
and try A on your own.
You don't need to graph it.
You can just solve it
and then check your solution.
So pause the movie
and give A a try.
[Silence] Ok,
let's see how you did.
Our U in this case,
is 5X plus 7.
So, 5X plus 7 is equal to 12
or 5X plus 7 is equal
to the opposite of 12.
So solving this equation,
we would get 5X is equal to 5
when I subtract 7
from both sides and then,
X would equal to one, or 5X,
when I subtract 7
from both sides,
will be negative 19.
So, dividing both sides by 5,
I would get X is
negative 19/5ths.
Now, let's go ahead
and check our solution.
[Pause] We'll do
that in a different color.
[ Silence ]
So, we'll check X equals 1
first and that would be
absolute 5 times 1 plus 7.
Is that equal to 12?
Well, 5 times 1 is 5.
5 plus 7 is 12.
So absolute value of 12,
is that equal to 12?
12 is 12, so that guy
checks out.
And then, for X equal
to negative 19/5ths,
we have 5,
absolute 5 times negative
19/5ths, plus 7.
Is that equal to 12?
These 5s divide out to be 1
so we would get negative 19
plus 7 is negative 12 inside
the absolute value.
Is that equal to 12?
12 does, indeed, equal to 12.
So it checks out.
So therefore,
our solution will be negative
19/5ths and 1.
[ Silence ]
Now, notice I'm solving this
using the algebraic processes
in rules for absolute
value equations.
I am not plugging
in random numbers
and checking.
So, checking at the end is ok
to make sure your answers
worked out, but I want you
to follow these processes.
I don't give any partial
credit for guess and check.
Ok. So, here we go.
Let's do Part B together.
Part B, we have [Pause] an
absolute value,
but it's being multiplied
by 7.
So, we first need
to get the absolute value
alone so you can either
multiply it by 1/7th,
which is equivalent
to dividing by 7.
So 1/7th times 7 is 1.
So, we'll end
up with absolute,
negative X plus 11 is equal
to 3.
And now we can apply the rules
for absolute value.
Negative X plus 11 is 3,
or negative X plus 11 is --
good, the opposite of 3.
So, negative X would equal
to negative 8.
X would be 8
or negative X would equal
to positive 8
and X would be negative 8.
[ Silence ]
Oh, my bad.
[ Silence ]
Negative X plus 11 would be
equal to negative 3.
When we subtract 11
from both sides,
we would get negative X is
negative 14
and multiplying both sides
by negative 1 gives us X
is 14.
[Pause] So now, come over here
and we do our check.
[Pause] So for X equal to 8,
we have 7 times absolute value
of negative 8 plus 11.
Is that equal to 21?
7 times absolute value of 3,
is that equal to 21?
7 times 3,
is that equal to 21?
21 equals 21.
It checks out.
Then, for X equal to 14,
we would have 7 times absolute
negative 14 plus 11.
Is that equal to 21?
7 times absolute
of negative 3.
Is that equal to 21?
[Pause] 7 times 3,
is that equal to 21?
21 equals 21
and it checks out.
So we get two results
that work.
We get 8 and 14.
[Pause] All right.
So moving right along,
we have another situation
where our absolute value is
not isolated so we have
to first add 8 to both sides
and we will get the absolute
value of X minus 4 equals 17.
And applying our rule,
we get X minus 4 is 17
or X minus 4 is negative 17.
[ Silence ]
So, X would be 21
or X would be negative 13.
[ Silence ]
And checking our result.
[ Silence ]
X equal to 21.
We would have absolute 21
minus 4.
Is that equal?
Oh. Sorry.
I've got to use the original.
Absolute 21 minus 4 minus 8,
is that equal to 9?
So we get absolute 17 minus 8.
Is that equal to 9?
17 minus 8.
And 9 is indeed, 9,
so that one checks out.
And X equal to negative 13.
We have absolute
that negative 13 minus 4
minus 8.
Is that equal to 9?
Absolute negative 17 minus 8,
is that equal to 9?
17 minus 8,
is that equal to 9?
9 equals 9 and you're good.
[Pause] And again, you know,
you can use your calculator
to check.
Since we have time
in the video,
I figured I would show
the checks.
So, our results were negative
13 and 21.
[Pause] Ok.
So, last up, again,
we have a situation
where the absolute value is
not isolated so we first need
to isolate absolute value of X
and we end up with negative 1.
Be careful,
you want to use your numbers
sense here.
This guy is an extraneous
[Pause] result, ok?
Remember, the absolute value
is the distance from zero.
Distance is not negative.
So if you get an absolute
value that's isolated
on one side
and then negative number
on the other side,
you end up with [Pause]
no solution.
So, remember,
you put either the empty set
brackets or you put the
empty set.
If you had gone through
and solved this equation,
what would have happened is
during the check,
you would have gotten answers
that didn't work.
All right.
[Pause] So, here we go,
moving along.
Now we move
on to [Pause] the case
where there's an absolute
value on each side.
All right?
So, if you have absolute U
equals absolute V,
you just pick one side
to designate as the, you know,
absolute value.
So you can say U is equal to V
or U is equal
to negative V. [Pause]
So here we go.
This one, using the definition
if U is to X minus 7
and V is the X minus 12,
we would say, 2X minus 7,
is equal to X minus 12
or 2X minus 7 is equal
to the opposite
or the negative of X minus 12.
And you've got
to put parenthesis around it.
So, solving, you see,
if we were to subtract an X
from both sides and add 7
to both sides,
we would get 1X is equal
to negative 5.
And over here, we need to work
through the right side a
little bit.
Distributing the minus,
we get negative X plus 12.
And now, if we were to add X
to both sides and add 7
to both sides,
we would have 3X is equal
to 19 and now isolating the X
we would get 19/3rds.
So again, checking
our results.
So, for X equal to negative 5,
we have absolute 2 times
negative 5 minus 7.
Is that equal to absolute?
Negative 5 minus 12.
So, simplifying inside the
parenthesis two times negative
5 is negative 10,
minus 7 is negative 17.
So we get absolute
negative 17.
Is that equal
to absolute negative 17?
Well, 17 equals 17
so that checks out.
Now for X equal to 19/3rds.
That will be a little bit
more computation.
Remember, you can use your
calculator for the check.
So absolute 2 times 19/3rds
minus 7, is that equal
to 19/3rds minus 12
in the absolute value?
[Pause] So here,
we'll end up with,
if we get a common
denominator,
we would have 38/3rds minus 21
all over 3.
Is that equal to, again,
common denominator,
we would have 19 minus 36 all
over 3.
[ Silence ]
So, 38 minus 27 gives us
17/3rds, or 38 minus 21,
rather, is 17.
It's all over 3
in the absolute value.
And then 19 minus 36 is
negative 17 and then,
that's over 3.
So, 17/3rds does, indeed,
equal to 17/3rds
so that one checks out.
So, our results will be
negative 5 and 19/3rds.