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# Worked example: absolute value equation with two solutions

Solving the equation 8|x+7|+4 = -6|x+7|+6 which has two possible solutions. Created by Sal Khan.

Video transcript

We're asked to solve for x. Let me just rewrite
this equation so that the absolute
values really pop out. So this is 8 times the
absolute value of x plus 7 plus 4-- in that same color--
is equal to negative 6 times the absolute value
of x plus 7 plus 6. Now the key here-- at first
it looks kind of daunting. It's this complex equation. You have these
absolute values in it. But the way to
think about this is if you could solve for the
absolute value expression, you could then-- it then turns
into a much simpler problem, then you can take it from there. So you could almost treat this
expression-- the absolute value of x plus 7, you can just
treat it as a variable, and then once you
solve for that, it becomes a simpler
absolute value problem. So let's try to do that. Let's try to solve
for not x first. We're just going to solve for
the absolute value of x plus 7. You'll see what I mean. So I want to get all of the
absolute values of x plus 7 on the left-hand side,
so I want to get rid of this one on the
right-hand side. Easiest way to get
rid of it is to add 6 times the absolute value of x
plus 7 to the right-hand side. We can't, of course, only do
that to the right-hand side. If these two things
are equal and we are being told
that they are, then if you add something
on this side, the only way that the
equality will hold is if you still do it
on the left-hand side. So let's do that, so plus 6
times the absolute value of x plus 7. And I want to get all
of these constant terms on to the right-hand side. So I want to get rid
of this positive 4. Easiest way is to subtract
4 right over there, but if we do it on
the left-hand side, we have to do it on the
right-hand side as well. And so what does this get us? So our left-hand side, if
I have 8 of something-- and in this case the something
is absolute values of x plus 7's-- but if I have 8 of
something and I add 6 of that same something, I now
have 14 of that something. So that's going to be
14 absolute values of x plus 7, 14 times the
absolute value of x plus 7. The 4 and the negative 4 cancel
out, and that was intentional. The negative 6 and the
6 x plus 7's cancel out, or absolute values of
x plus 7's cancel out, and that was intentional. And then we're left with 6
minus 4, which is just 2. So that's going
to be equal to 2. Now just as promised,
we want to solve for the absolute
value of x plus 7, so let's divide both
sides by 14 to get rid of that coefficient
there, or that factor, or whatever you want to
call it, the thing that's multiplying the absolute
value of x plus 7. So we'll divide
both sides by 14, and we are left with
the absolute value of x plus 7 is equal to 2/14. They're both divisible by 2, so
this is the same thing as 1/7. So just as promised,
we've now solved for the absolute
value of x plus 7, but we really need
to solve for x. So how can we
reason through this? So if I take the absolute value
of something and I got you 1/7, there's two possible things that
I took the absolute value of. I could have taken the
absolute value of positive 1/7, or I could've taken the
absolute value of negative 1/7. So this thing that we're
taking the absolute value of-- so x plus 7-- could
be equal to positive 1/7, or x plus 7 could be
equal to negative 1/7. And just think about
that for a second. If this thing right over
here were equal to 1/7, you take its absolute
value, it'd be 1/7. If this thing was negative 1/7,
you take its absolute value, it would be positive 1/7. So that's how we got this. So now let's just solve for x. So if we subtract
7 from both sides for this left-hand
equation, we get x is equal to 1/7
minus-- and 7 we can rewrite as 49/7, which
is equal to negative 48/7. So that's one possibility for x. And then the other
possibility we would get x is equal to-- so we
have negative 1/7 minus 49/7. We're just subtracting
7 from both sides. That's what 49/7 is. And then this gets
us to negative 50/7. So the two solutions
to this what we thought was a complicated
equation are negative 48/7 and negative 50/7.