- [Voiceover] Perform the indicated operations,
then simplify.
So we have three rational expressions here,
and we're gonna subtract this one,
and we're going to add this one.
So just like if we were adding
and subtracting a bunch of fractions,
we really have to find a common denominator here.
And to figure out a common denominator,
we have to figure out the least common multiple
of x plus five, of 2x minus 10,
and x-squared minus 25.
And to find the least common multiple,
we really want to factor these out
and make sure that that least common multiple,
that that common denominator,
has all of the constituent factors in it.
So this can't be factored anymore.
X plus five, that's about as simple as you can get it.
2x minus 10.
We can factor out a two, and it becomes
two times x minus 5.
So we factor that out.
And then x-squared minus 25
you might recognize as a difference of squares,
so that's the same thing as
x plus five times x minus five.
Just factored that out.
And so if we want to have a least common multiple,
so let me rewrite each of these
with that common denominator here,
that the least common multiple,
the least common multiple
is going to have to be something that has x plus five in it,
it's gonna have to have a two in it,
it's gonna have an x minus five in it,
and we've already said it has to have an
x plus five and an x minus five in it.
So it's gotta have a two.
It's gotta have a two.
It's gotta have an x plus five.
It's gotta have an x minus five in it.
And it's gotta have an x plus five times an x minus five.
Well, we've already got that there,
so no need to write it again.
So that right there is our common denominator,
so let's write it out.
So let's rewrite each of these expressions
with that common denominator.
So two times x plus five
times x minus five.
And then right over here I'll write it again.
Two times x plus five
times x minus five.
Once you get used to this,
you might be able to just write out the common denominator
once and then figure out the appropriate
things to multiply the numerators by,
but I want to make sure we go through every step
for these videos just so we make sure
we understand what's going on.
So to go from x plus five to two times x plus five
times x minus five.
What do we have to multiply x plus five by?
Well, we have to multiply it in order to get at this.
We're multiplying it by a two.
We're multiplying the x plus five by a two,
and we're also multiplying it by
an x minus five.
So if we're multiplying the denominator by a two
and an x minus five,
we have multiply the numerator by a two
and an x minus five.
So if we multiply two times x minus five times one,
we'll get two times x minus five,
and we don't even have to write this one.
Now we're subtracting over here.
And to get from two times x minus five to this over here,
what to we have to multiply it by?
Well, we have the two times x minus five right over there.
We have to multiply it by an x plus five.
This is the part that is new.
We have to multiply it by an x plus five.
We have to multiply this denominator right here
by x plus five to get this denominator.
So if you multiply the denominator by x plus five,
you want to multiply the numerator by x plus five as well.
So the numerator's going to be two times x plus five.
Just to put two, multiplied it by x plus five.
Exactly what we did to the denominator.
We multiplied it by x plus five.
Then we have this last expression right over here.
I'm going to do this in a new color.
We went from x plus five times x minus five
to x plus five times x minus five
times two.
So to go from this denominator to this denominator
we just have to multiply it by a two.
If you multiplied the denominator by two,
you should multiply the numerator by two.
So multiply two times 2x.
You're going to get 4x.
So all I did here is I multiplied it by two.
Let me make it clear.
To get from this to this,
I multiplied it by x plus five.
X plus five.
And then to go from there to there,
I multiplied it by two times x minus five.
And over here I should use,
instead of writing multiplied by two right there,
I should multiply it by two.
Don't want to get the multiplication symbol confused
with the x variable.
And of course, there was a positive out here.
So we're now ready to do our adding and subtracting.
And we could view this just for simplicity,
just make it clear.
Subtracting this thing is the same thing
as adding the negative of it.
And that makes it a little bit simpler.
So let's put our common denominator down.
Let's put our common denominator down.
So all of this business is going to be,
and our denominator's going to be two times x plus five
times x minus five.
And our numerator is now going to be,
and I'm going to multiply this out,
so that we can properly add terms.
So two times x minus five is 2x minus 10.
And then we have a negative two times x plus five,
so that's going to be negative 2x,
and then negative two times five is negative 10.
And then we're going to have a plus 4x.
Plus 4x over here.
Plus 4x, and then we can simplify this thing.
And so you have a 2x.
You're subtracting a 2x.
Those guys cancel out.
And then you're left with just a 4x.
So you're left with just a 4x,
and then minus 10 minus 10.
So 4x minus 20
over two times x plus five
times x minus five.
Let's see, we can factor out a four up here.
So if we factor out a four,
this becomes four times x minus five.
If we assume that we're not dealing
with x is equal to positive five,
'cause that would make the denominator undefined,
we can divide the numerator and the denominator
by x minus five.
And so if we divide the numerator and the denominator
by x minus five, those cancel out.
And we can divide the numerator and the denominator by two,
since they're both divisible by two.
That four divided by two is two.
Two divided by two is one.
So the numerator, we're just left with a two,
and in the denominator, we're just left with an x plus five.
We're just left with an x plus five.
Now, I want to be very careful here,
because this expression up here,
if we really want this final answer that we got
to be truly identical to this expression over here,
just a simplified version,
we have to make sure that it has the same domain.
If say this was part of the function definition,
you want to make sure that it's valid
for the same x's and not any more,
if you want them to be truly identical.
And this up here is not valid.
So the domain if you viewed this as one entire expression,
or one entire function definition,
the domain would be
all real x's,
except
x cannot be equal to negative five,
or x cannot be equal to positive five,
'cause that would make this denominator equal zero,
or x is equal to five.
And so if you want these two things to be identical,
they have to have the same domain.
Most algebra classes,
people will not be particular about this.
And maybe you don't want them to be the same function.
Maybe you want it to be defined
when x is equal to a positive five.
But if you want them to be truly the same expression,
you have to have the same domain.
So your domain you would want to say
it is all real
x's except...
If you were just to do the domain for this thing,
if you just solved this expression right here and say
okay, x can't be equal to negative five.
So let me write except x is equal to negative five.
And so this right here is just a domain for this thing,
but if we wanted to have the same domain
as the original expression,
we'd also throw in and x is equal to five.