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# Worked example: absolute value equations with no solution

Solving the equation 4|x+10|+4 = 6|x+10|+10 to find that it has no possible solution. Created by Sal Khan.

Video transcript

So we're asked to
solve for x, and we have this equation with
absolute values in it. So it's 4 times
the absolute value of x plus 10 plus 4 is equal to
6 times the absolute value of x plus 10 plus 10. And at first, this
looks really daunting, but the key is to just solve for
this absolute value expression and then go from there. Let me just rewrite it so
that the absolute value expression really jumps out. So this is 4 times
the absolute value of x plus 10 plus 4 is equal to
6 times the absolute value of x plus 10 plus 10. So let's get all of the
absolute values of x plus 10 on the left-hand side. So I want to get
rid of the 6 times the absolute value of
x plus 10 on the right. Well, how would I do that? Well, I could subtract 6 times
the absolute value of x plus 10 from the right,
but we've already seen this multiple times. If these two things
are equal, and if I want to keep them
equal, if I subtract 6 from the right-hand side,
I've got to subtract-- or if I subtract 6 times the
absolute value of x plus 10 from the right-hand side, I
have to subtract the same thing from the left-hand side. So we're going to
have minus 6 times the absolute value of x plus 10. And likewise, I want to
get all my constant terms, I want to get this 4 out
of the left-hand side. So let me subtract
4 from the left, and then I have to also
do it on the right, otherwise my equality
wouldn't hold. And now let's see
what we end up with. So on the left-hand side,
the 4 minus 4, that's 0. You have 4 of something
minus 6 of something, that means you're going
to end up with negative 2 of that something. Negative 2 of the absolute
value of x plus 10. Remember, this might
seem a little confusing, but remember, if
you had 4 apples and you subtract 6 apples, you
now have negative 2 apples, I guess you owe
someone the apples. Same way, you have 4
of this expression, you take away 6 of
this expression, you now have negative
2 of this expression. Let me write it a
little bit neater. So it's negative 2 times the
absolute value of x plus 10 is equal to, well the whole
point of this, of the 6 times the absolute value of
x plus 10 minus 6 times the absolute value of x plus
10 is to make those cancel out, and then you have 10 minus
4, which is equal to 6. Now, we want to solve for the
absolute value of x plus 10. So let's get rid
of this negative 2, and we can do that by dividing
both sides by negative 2. You might realize,
everything we've done so far is just treating
this red expression as almost just like
a variable, and we're going to solve for
that red expression and then take it from there. So negative 2 divided
by negative 2 is 1. 6 divided by negative
2 is negative 3. So we get the absolute
value of x plus 10 is equal to negative 3. Now, this gets us to a
very interesting situation. You might say maybe this
could be the positive version or the negative, but
remember, absolute value is always non-negative. If you took the absolute
value of 0, you would get 0. But the absolute
value of anything else is going to be positive. So this thing right
over here is definitely going to be greater
than or equal to 0. Doesn't matter what
x you put in there, when you take its
absolute value, you're going to
get a value that's greater than or equal to 0. So there's no x that you
could find that's somehow-- you put it there, you add 10,
you take the absolute value of it, you're actually
getting a negative value. So this right over here
has absolutely no solution. And I'll put some exclamation
marks there for emphasis.