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Studying for a test? Prepare with these 3 lessons on Absolute value equations, functions, & inequalities.

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# Worked example: absolute value equations with one solution

Video transcript

We're told, solve the absolute
value of 3x minus 9 is equal to 0, and graph the solution
on a number line. So let's just rewrite the
absolute value equation. They told us that the absolute
value of 3x minus 9 is equal to 0. So we're told that the
absolute value of the something-- in this case the
something is 3x minus 9-- is equal to 0. If I told you that the absolute
value of something is equal to 0, I'm telling you
that something has to be exactly 0 away from 0, or 0 away
from the origin on the number line. So the only thing that that
something could be is 0. If I told you that the absolute
value of x is equal to 0, you know that x has
to be equal to 0. That's the only value whose
absolute value is 0. So if I told you that the
absolute value of 3x minus 9 is 0, than we know that 3x minus
9 has to be equal to 0, and that's kind of unique about
the 0 is that, it's the only number the has a unique,
that's only the absolute value of 0. If you had, say, a 1 here, you
could say, oh well, then this thing could be a 1
or a negative 1. But here, if you have a 0,
this thing can only be 0. So solving this equation is
fairly straightforward. If we want to isolate the 3x,
get rid of the negative 9 on the left-hand side, we add 9 to
both sides of the equation. Add 9 to both sides
of the equation, these 9's cancel out. That's the whole point. On the left-hand side, you're
just left with 3x, and on the right-hand side, you are
just left with 9. Now we want to solve for x,
so we have 3 times x. Let's divide it by 3, because 3
times x divided by 3 is just going to be x. But if we divide the left side
by 3, we have to divide the right side by 3. So we are left with-- these
guys cancel out. x is equal to 9 over
3, which is 3. And that's our solution. Now let's try out. Let's make sure that this
actually works. Let's substitute it back into
our original equation. So we have the absolute
value of 3 times x. Instead of x, I'll just put in
our actual answer that we got, 3 times 3 minus 9 has got
to be equal to 0. So what's this going
to be equal to? 3 times 3 is 9. So it's the absolute value of
9 minus 9, which is the absolute value of 0, which
is, indeed, 0. So it does, indeed, equal
0, and we are done.