BAM!!! Mr. Tarrou In this lesson we are going
to look at solving equations with the absolute
value function in it...and inequalities. This
video is intended for students in algebra
1 and beginning algebra 2. So we are not going
to be taking absolute value functions and
wrapping them around any crazy functions like
sine, or cosine, or absolute values wrapped
around the equation of a parabola. We are
not going to get that complex. But we do need
to have a starting point. How you solve an
equation or inequality with an absolute value
symbol. Well, you isolate the absolute value
function. You rewrite the equation without
the absolute value function by separating
it into two separate parts. This first part
you will just drop the absolute value function.
The second part you replace the absolute values
with a set of negative parenthesis. Now you
have to separate these into two parts because
the absolute value function says or explains/describes
how far a value is from zero on the number
line. So if I said, "What values are three
units away from zero?" well three and negative
three. Three is three units to the right and
negative three also three units away, but
it is three units to the left. So both three
and negative three are three units away from
the center of the number line or zero. So,
absolute value functions will normally have
two answers and certainly when we are solving
them we need to break that absolute value
equation or inequality into two parts to account
for those left and right hand side values
on the number line. So, we will be doing that
in a second with some examples. Solve each
equation, each of the two you have from separation
from step two. And then check your answers
to make sure that what you got is actually
going to work with your absolute value function.
So lets see what this is going to look like.
We will start off with a really simple example
and then step it up a little bit in difficulty
as we go. We will also compare the equations
with inequality problems. Number one. The
absolute value of x equals 8. Clearly there
is no work here. The absolute value function
is already alone and there really is nothing
interesting inside the absolute function.
We are just showing you how to pull apart
the equation into two parts and also show
you why you want to keep one positive and
one negative. So, the absolute value of x
equals eight. Well, I am just going to write
x equal to eight writing the equation without
the absolute value function. Now I am going
to replace the absolute value function with
a set of negative parentheses. So we have
x equals 8 and negative x equals eight. We
are going to need to divide both sides by
negative one and we get x is equal to negative
eight. Let's check that answer. Well, is the
absolute value of eight equal to eight? Yes.
And is the absolute value of negative eight
equal to eight? The absolute value of negative
eight is eight and that indeed works out as
well. A VERY simple example, you may even
say too simple but I am showing you that I
am writing the equation with no absolute value
symbols at all and I get a solution that works.
HaHa, I am not sure why I have this huge empty
space here. Then I took out the absolute value
symbol and I replaced it with a set of negative
parenthesis. Now this is a little bit of a
different format than my textbook and other
textbooks use. I like writing it like this
where I say that you take the absolute values
out and replace them with a set of negative
parenthesis because of working with inequalities.
Now what I mean by that is, if I go up here
and replace the equal sign with lets say a
less than sign. Now we have x is less than
8. And by taking out the absolute value function
and replacing it with a set of negative parenthesis
you are going to see why maybe your textbook
says... It doesn't explain how to do your
problem this way but with inequalities it
will just magically say change the direction
of the inequality and I am going to divide
both sides by negative one. Now we get x is
greater than negative eight. See a lot of
books will show when you break apart an absolute
value function that you write the inequality
in its given format except drop the absolute
value. And then with the other inequality
you change the direction of the inequality
and put the negative on the other side. Some
students will not understand completely why
the inequality changes direction. So, that
is why I teach this the way that I teach it.
OH, number line. so the absolute value of
x is less that 8 on the number line is going
to look like this. We have zero, we have negative
eight, and we have eight. I did not include
an equal sign so this is going to be an open
dot on 8 and shade to the left. This is going
to be on open dot on negative eight and shade
to the right. Thus we have in this example
shading between negative eight and eight.
What would happen if we changed the direction
of the inequality and my example was the absolute
value of x to be greater than eight. We have
x is greater than 8. Negative x is greater
than 8. Now divide both sides by negative
one and we get x is less than negative eight.
Now when I graph this on a number line. I
have my zero, my eight, and my negative eight.
I have an open dot on eight and I shade to
the right. I have an open dot on negative
eight and I shade to the left. So as long
as I use a number that is less than negative
eight or greater than positive eight that
will be a solution to that absolute value
inequality. There is three examples but just
with x and 8. Not very much work there. So
lets get into some meaty questions here, some
that require a bit more work. How about the
absolute value of 3x minus 1 is equal to fourteen.
My absolute value function is already isolated
so I am going to write 3x minus one is equal
to fourteen. Take the absolute values out
and replace them with a set of negative parenthesis.
So, negative times three x minus one equals
fourteen. And then solve each equation independently
of each other. Add one to both sides and get
three x is equal to fifteen. Divide by three
to undo that multiplication and you get x
equals five. Then over here you have two choices.
You can distribute the negative one through
the parenthesis or you can divide both sides
by negative one to get rid of that multiplication
of negative one. That negative in front of
the parenthesis acts like a negative one that
wants to get multiplied through the parenthesis
because it is negative one times three x minus
one. So we are going to cancel that out and
get three x minus one equals negative fourteen.
We are going to undo that subtraction of one
by adding both sides by one. When ever you
are trying to get something over the equal
sign you want to use the inverse math operation...or
the opposite. Negative fourteen plus one is
negative thirteen. Divide both sides by three
and we get x is equal to negative thirteen
over three. Now sometimes you get solutions
that don't actually work so you want to check
that. So let's take the first answer. The
absolute value of three times five minus one
equals fourteen. That is going to be the absolute
value of fifteen minus one is equal to fourteen.
That is the absolute value of fourteen equals
fourteen and indeed that does work. Our first
solution is good, let's try the next one.
The absolute value of three times negative
thirteen over three minus one equals fourteen.
These threes cancel out or you can do three
times negative thirteen is negative thirty-nine
divided by three minus one. This divides,
thirty-nine divided by three is negative thirteen
minus one. That comes out to be negative fourteen
and the absolute value of negative fourteen
is fourteen so we have a check there as well.
So we had two solutions and they both checked
out. If you were not given this problem as
an equation but if you were given this problem
as an inequality, how would the work change?
Well it would not change by very much. If
I want the absolute value of 3x minus 1 to
be greater than or equal to 14, solving inequalities
is exactly the same as solving equations.
EXCEPT FOR when you divide or multiply by
a negative number. When you divide or multiply
both sides of an inequality by a negative
number you are going to need to remember to
change the direction of that inequality. Let's
put that on a number line and then move on
to our next example. Here is our number line.
We have zero, one, two, three, four, five,
and we are going to have a closed dot on five
because of the equal sign. It is x is greater
than or equal to five so we are going to shade
to the right. Where in the heck is negative
thirteen over three? You might want to make
this into an improper fraction before you
bother trying to put it on the number line.
Thirteen divided by three. How many threes
are in thirteen? 3, 6, 9, 12. So this is going
to be x is less than or equal to negative
four... and then three times four is twelve
giving you a remainder of one. So that is
negative four and one third. So negative one,
negative two, negative three, negative four,
negative five. Negative four and one third
take you just a little bit beyond the negative
four on the number line. It is x is less than
or equal to and x's get smaller as you move
to the left. There is the shading of this
solution. Next example! ALRIGGGGHHHHHTYYYY
THEN! We have negative two time the absolute
value of x minus four plus six equals ten.
This absolute value function is not already
alone so we are going to have to take a couple
of steps before we are ready to start this
problem. We are going to subtract both sides
by six. You always want to add or subtract
away from the variable. And in this case the
variable is inside this math function and
then take care of what is touching last. So
negative two times the absolute value of x
minus four equal to four. We are going to
divide both sides by negative two. We get
the absolute value of x minus four equals
negative two. We now have the absolute value
isolated. But I don't really need to continue
the problem. If you do you are going to eventually
come to the same conclusion when you try to
check your solutions. And you may not notice
you made a mistake unless you do take those
answers and try to plug them back into the
equation and see if they work. Because you
can continue this problem. Nothing will seem
to be wrong. You will get those solutions
when you are done and if you the time to try
and plug them back into the original absolute
value function you are going to realize that
none of your answers work. If you don't check
you will never realize that and get a lot
of points marked off for that question. The
absolute value of a number tells you how far
that number is away from zero on the number
line. There really no such thing as a negative
distance. This is saying the absolutely value
of something is negative two. Well the absolute
value of two is two and the absolute value
of negative two is two. An absolute value
will never give you a negative answer as a
result. This right here. The fact that we
have negative two means that there is no solution.
So if you continue on with this problem and
you find out what you think are values of
x when you are done and you don't check solutions
you are not going to know that you made a
mistake and you should have stopped way back
here. But this is no solution. Absolute values
do not give us negative numbers as a final
answer. Moving on to the next one. The absolute
value of 2x plus five is equal to 3x plus
four. Ok, the absolute value function is already
isolated so we are going to break this up
into two parts. So we have 2x plus 5 is equal
to 3x plus 4 and we taking out the absolute
value and replacing it with a set of negative
parenthesis, negative 2x plus 5 is equal to
3x plus 4. Ok we have variables on both sides
of this equation so I can move anything I
like. I like to avoid creating negative numbers
so I am going to subtract both sides by 2x
and get 5 equals x plus 4. Subtract away from
the variable which I have put on the right
hand side. You don't see that very often,
so sometimes now students will start to make
a mistake because the variable is on the right
when they are used to seeing it on the left.
You want to move away from the variable and
we have a plus 4 with the x. So we are going
to subtract both sides by 4 and get x equals
1. Over here, just for the heck of it I am
going to distribute the negative one through
the parenthesis instead of dividing it to
the other side. Why would I, I have two terms
over there to deal with as well so I would
not be saving any work. We have got negative
one times two which is negative two, now it
is negative five equals 3x plus four. I am
going to again, just to avoid some negative
work, I am going to take this negative two
x and add it to the other side. We get negative
five equals five x plus four. Subtract both
sides by four, get negative nine equals five
x. Now undo this multiplication of five by
dividing by five, again doing that inverse
math operation when ever you are trying to
get away from the variable. We get our other
solution of x equals negative nine fifths.
So all the algebra is good. We have a couple
of answers that hopefully are going to work,
but if we do not check them we are not going
to be sure. So, we have got... Lets try number
one. The absolute value of 2 times 1 plus
5 equals 3 times 1 plus 4. Do you see what
I am doing? I am just taking the value of
x which we found, the value of 1, and plugging
it where we used to have the x variables.
So 2 times 1 is 2. 2 plus 5 is 7. The absolute
value of 7 is 7. Over here we have 3 times
1 plus 4 which is also equal to 7 and so our
first solution of x equals 1 worked. That
is good! Let's try our other solution of -9/5.
Where we did have a 1 in for x we are going
to put in -9/5. We have got 2 over... let's
make that 2 look like a fraction so we can
see that we are going to multiply the two
numerators and two denominators. So two times
negative nine is negative eighteen and one
times five which is five plus five. Now I
am going to have to add these two terms if
I am going to finish checking my answer. I
don't want to just be dependent on a scientific
calculator, so I am going to write my five
as a fraction. This way I can find the common
denominators. I cannot add fractions if I
do not have common denominators. So I am going
to multiply this on the top and bottom by
five and get the absolute value of 18/5 plus
25/5. That is going to come out to be negative
eighteen plus twenty five is seven over five.
And the absolute value of seven fifths is
seven fifths. Ok, let's try the other side
and make sure that it is the same value. We
have three times negative nine which is equal
to negative twenty seven over five because
that three is over one. Right, just multiply
the numerators and then one times five is
five plus four. Well again, you cannot add
fractions unless you have common denominators
so stick that four over one. We are going
to multiply that second fraction and the top
and bottom by five to get that common denominator.
So we have negative twenty seven plus twenty
over out common denominator of five. We can
see what is going to happen here. Negative
twenty seven plus twenty is negative seven
over five. These are not equal. So this solution
is no good or extraneous. So the algebra showed
two solutions of one and negative nine fifths,
but the negative nine fifths does not work
and you will know that unless you check your
answers. Ok, one last example. Lets through
something else in with a little bit of fraction
work. The absolute value of x minus four over
two minus one is less than five. The last
two were equations so lets stick an inequality
in there. You cannot break up an inequality
unless....excuse me an absolute value unless
it is isolated. So, I am going to subtract
both sides by one to undo that addition of
one. You get the absolute value of x minus
four over two is less that four. We are now
going to break this up into two parts because
now our absolute value function is alone.
We get x minus four over two is less than
four and we get negative x minus four over
two is less that four. We are going to go
ahead and undo.... We haven't had an equation
yet with a fraction bar and students generally
hate fractions because they just take a little
bit more time to work. We are so dependent
on our calculators any more that people just
forget how to use them. A fraction bar does
mean division, yes, so this is x minus four
divided by two. If I want I want to get two
away from the x I want to undo that division
of 2 and the inverse of division is multiplication.
I am going to multiply both sides of my equation
by two to undo that division of two. They
are simply going to cancel out, cross out.
So we have two divided by two, the multiplication
and the division of two are going to cancel
out. The fraction bar also acts as one big
grouping symbol so I do not have to worry
about the fact that there is two terms on
top. They are being grouped together by the
fraction bar. The two's cancel out and you
get x minus four is less that eight. Don't
change the direction of that inequality because
I multiplied by a positive number, not a negative
one. Add four to both sides and get x is less
than twelve. Over here I am going to divide
both sides by negative one. I don't want to
take that negative, I could, but I don't want
to take that negative one that is in front
of the parenthesis and distribute it through.
It is less work to just simply divide over
to the other side. And that process is going
to, as you should, change the direction of
the inequality because I divided by a negative
number thus the inequality becomes greater
than. Undo the division of two by multiplying
by two again. That is going to cancel and
we get x minus four is greater than negative
eight. Add both sides by four to undo that
subtraction of four. We get x is greater than
negative four. Since this is an equality lets
go ahead and graph this on a number line.
So we have zero. Let's just do a quick sketch.
We have negative four and we have twelve.
No equal signs so these are open dots. X is
less that 12 so an open dot on twelve and
x values which are less that 12 are to the
left. X is great than negative four so an
open dot again because there is no equal sign.
It says x is greater than and your x's get
bigger to the right. This is my shading for
the last example. I am Mr. Tarrou. BAM!!!
Go Do Your Homework!
