All right, in this video we're going to
go over solving absolute value equations
and we'll do
four different kinds we'll start with
sort of an easier one like this
and then we'll work our way up so the
idea is that if you have an absolute
value
what you're going to want to do is set
the argument there the
the expression on the inside of the
absolute value
equal to plus or minus that number all
right so when we do that we get
7x minus 4 equals negative 10
or 7x minus 4 equals
positive 10 like that okay so once we do
this
we have two equations that we can solve
pretty easily just using regular algebra
right so we would add 4 here and that
would leave us with
7x equals negative 6
okay dividing by 7 we get
one solution at negative 6 7
and then solving this piece right we'll
add 4 on both sides so we get
7x equals 14 right and then dividing by
7 gives us our second solution here
x equals 2. now you can go ahead and
take those plug it in see that they
really do solve the original
it's pretty clear that it does but let's
just go ahead and move on we're going to
do a little more complicated example
next
okay so now you can see here we have an
absolute value equation we have an
absolute value in the equation but it's
not
isolated so the first step in solving
these things is to actually isolate the
absolute value
okay so to do that what we're going to
do is
use a little bit of algebra here we'll
go ahead and subtract 5 on both sides
and that'll leave us with negative
absolute value
3x equals negative 12.
okay now from here we have a negative in
front of the absolute value there's a
couple ways to deal with that
we can either multiply both sides by
negative one
or in this case i'm just going to divide
both sides by negative 1
and negative 1 divided by negative 1
cancel leaving me with
absolute value of 3x equals positive 12.
all right so that's the first step you
have to isolate the absolute value
once you get the absolute value isolated
then you set the argument equal to plus
or minus that number
all right so we get 3x equals negative
12
or 3x equals positive 12.
okay dividing by 3 here x then equals
negative 4
and then dividing by 3 on that one x
could equal positive 4. and again you
could plug those back into the original
and see that they really do work
okay now let's do another one all right
so let's look at this one
this one says the absolute value of x
minus three equals
x plus one so the absolute value is
isolated but now
the difference is that it's equal to a
quantity not just a number
well the idea for solving this is the
same so whatever's inside the absolute
value we're going to set it equal to
plus or minus
that in this particular case we have
x minus 3 equals the opposite
of that quantity x plus 1 so minus
or x minus 3 equals
positive that quantity
okay so once we're here we just do the
algebra
we don't we no longer have an absolute
value so we can just do regular algebra
so we have x minus 3
equals negative x minus 1 distributing
the negative there
and then adding x
right that'll give me 2x minus 3 equals
negative 1.
2x equals i'm going to add 3 2x equals 2
and then x equals 1 after i divide by 2
will be the first solution that i get
there all right and then over here we
have
x minus 3 equals x plus 1.
now when i subtract x like i did on the
previous example
i'm left with well the x's add to zero
so i'm left with negative three equals
one and so that's a false statement
and we know in math when we get a false
statement
this piece contributes no solution
so the only solution that we got out of
this
calculation is 1. and you can actually
plug that in see
1 minus 3 is negative 2 an absolute
value
equals 1 plus 1 which is 2. so 2 equals
2 it checks out
all right let's do one more type of
example you're going to run into
all right so here we have an absolute
value equals an absolute value
sometimes that throws people off but
we're going to work it the same way
basically we're going to set this
argument x minus 1 equal to plus or
minus that argument
it doesn't matter which way you do it
you'll get the same results
so in this particular case i have x
minus 5
equals the opposite here so
negative 2x minus 1 or
the x minus 5 is going to equal positive
right
2x minus 1. and then we can solve each
one of these little quantities with
regular algebra
so the first step here would be to of
course distribute the negative
and we get x minus 5 equals negative 2x
plus 1
and then adding 2x to both sides
that'll leave us with three x minus five
equals one adding five
three x equals six and then x equals
two and you can actually plug that in
check to see if it works
here we have actually x minus five
equals two x
minus one so we'll go ahead and subtract
two x x minus two x is negative x
so negative x minus five equals negative
one
adding five negative x equals
four and then multiplying or dividing by
negative one
x equals negative four okay so those are
my
two solutions when we had an absolute
value equals an absolute value
now at the beginning of the video i said
there was four examples
but there's actually one more type if
you stick around it'll be worth it
we'll do one more as a bonus question
okay so here it is
the difference is that we have the
absolute value of x squared minus five
equals four the difference is that we
have a quadratic
inside the absolute value okay the steps
for solving are going to be the same
but the algebra steps after we get out
of the absolute value notation are going
to be different
all right so let's see what we have we
have x squared minus 5 well that's going
to equal negative 4
or x squared minus 5 is going to equal
positive 4 right so that's the same
theorem that we're using
for the absolute value now to solve this
quadratic there's a number of ways to go
since there's no middle term x i'm going
to solve by extracting square roots
i think it'll be easiest if i just go
ahead and add 5 to both sides here
and i get x squared equals 1.
all right and then we'll take the square
root of both sides don't forget when you
take the square root of both sides
let me have a little plus or minus going
on
so the square root of x squared we get x
equals plus or minus the square root of
one which is just
one all right so this piece gives me two
solutions for that
absolute value inequality now we'll go
ahead and do the same thing here adding
5
well that gives us a 9 on the right side
so we have x squared
equals 9 all right and then taking the
square root of both sides
not forgetting the plus or minus again
of course
we have x equals plus or minus the
square root of nine which is
three so you can see in this case we had
an
absolute value with a quadratic in it
and when we did the steps we came out to
four answers if we were to try those
they all work
okay so there you go five different
types of absolute value equations that
you'll run into
so good luck
