Adding and subtracting three fractions

We have negative 3/4
minus 7/6 minus 3/6.

And there's many
ways to do this.

But it immediately
jumps out at me

that these last two numbers
have a 6 in the denominator.

So I'm going to worry
about these first.

I'm going to view this as
negative 7/6 minus 3/6.

So if we have negative
7/6 minus 3/6,

that's going to be the same
thing as negative 7 minus 3

over 6.

And of course, we
have this negative 3/4

out front that
we're going to add

to whatever we get over here.

So this is these two terms
that I'm adding together.

Negative 7 minus
3 is negative 10.

So it's negative 10 over 6.

And then I'm going to have
to add that to negative 3/4.

And now I have to worry about
finding a common denominator.

Let me write that so
they have a similar size.

So now I have to worry about
finding a common denominator.

What is the smallest number that
is a multiple of both 4 and 6?

Well, it might jump out
at you that it's 12.

You can literally just go
through the multiples of 4.

Or you could look at
the prime factorization

of both of these numbers.

And what's the
smallest number that

has all of the prime
factors of both of these?

So you need two 2s, and
you need a 2 and a 3.

So if you have two 2s and a
3, that's 4 times 3 is 12.

So let's rewrite this
as something over 12

plus something over 12.

Well, to get your
denominator from 4 to 12,

you have to multiply by 3.

So let's multiply our
numerator by 3 as well.

So if we multiply
negative 3 times 3,

you're going to have negative 9.

And to get your
denominator from 6 to 12,

you have to multiply by 2.

So let's multiply our
numerator by 2 as well so

that we don't change the
value of the fraction.

So that's going
to be negative 20.

And now we're ready to add.

Our common denominator is 12.

And so this is going to be
negative 9 plus negative 20,

or we could even write
that as minus 20,

over 12, which is equal to--
and we deserve a drum roll now.

This is negative 29 over 12.

And 29 is a prime
number, so it's not

going to share any common
factors other than 1 with 12.

So we also have this in
the most simplified form.