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CCSS.Math:

Simplify the rational
expression and state the domain. Let's see if we can start with
the domain part of the question, if we can start
with stating the domain. Now, the domain is the set of
all of the x values that you can legitimately input into
this if you view this as a function, if you said this is
f of x is equal to that. The domain is a set of all x
values that you could input into this function and
get something that is well-defined. The one x value that would make
this undefined is the x value that would make the
denominator equal 0-- the x value that would make
that equal 0. So when does that happen? Six minus x is equal to 0. Let's add x to both sides. We get 6 is equal to x, so the
domain of this function is equal to the set of all
real numbers except 6. So, x could be all real numbers
except 6, because if x is 6 then you're dividing
by 0, and then this expression is undefined. We've stated the domain, now
let's do the simplifying the rational expression. Let me rewrite it over here. We have x squared minus
36 over 6 minus x. Now, this might jump out at you
immediately, as it's that special type of binomial. It's of the form a squared minus
b squared, and we've seen this multiple times. This is equivalent to a plus
b times a minus b. And in this case, a
is x and b is 6. This top expression right here
can be factored as x plus 6 times x minus 6, all of
that over 6 minus x. Now, at first you might say,
I have a x minus 6 and a 6 minus x. Those aren't quite equal, but
what maybe will jump out at you is that these are the
negatives of each other. Try it out. Let's multiply by negative 1 and
then by negative 1 again. Think of it that way. If I multiply by negative 1
times negative 1, obviously, I'm just multiplying the
numerator by 1, so I'm not in any way changing
the numerator. What happens if we just multiply
the x minus 6 by that first negative 1? What happens to that
x minus 6? Let me rewrite the
whole expression. We have x plus 6, and I'm
going to distribute this negative 1. If I distribute the negative 1,
I have negative 1 times x is negative x. Negative 1 times negative
6 is plus 6. And then I have a negative
1 out here. I have a negative 1 times
negative 1, and all of that is over 6 minus x. Now, negative plus 6. This is the exact same thing
as 6 minus x if you just rearrange the two terms.
Negative x plus 6 is the same thing as 6 plus negative
x, or 6 minus x. Now you could cancel them out. 6 minus x divided by 6 minus x,
and all you're left with is a negative 1-- I'll write it
out front-- times x plus 6. If you want, you can distribute
it and you would get negative x minus 6. That's the simplified
rational expression. In general, you don't have to
go through this exercise, multiplying by a negative
1 and a negative 1. But you should always be able
recognize that if you have a minus b over b minus a that that
is equal to negative 1. Or think of it this way: a
minus b is equal to the negative of b minus a. If you distribute this negative
sign, you get negative b plus a, which
is exactly what this is over here. We're all done.