In this exercise we’re asked to subtract

the rational expressions 3 over x minus six minus x plus 15 over x minus 6 times

x plus 1. Remember that rational expressions are just big fractions, so

just like with adding and subtracting fractions, you need a lowest common

denominator. Well in order to get a common denominator, the first step with

rational expressions is to factor each of the denominators. It’s already done

for us here – you have x minus 6 times the quantity x plus 1 in the second fraction,

or rational expression. And this first one is x minus 6 – you can’t factor that

anymore to get a common denominator. Think about building up the denominators;

that is what is this missing what factor is this missing that this one has? Well

it’s x plus 1 so we’re going to multiply both the numerator and the denominator

by x plus 1 just like you would multiply the numerator and the denominator by

something when you were adding or subtracting fractions. Now we’re going to

multiply the 3 times the x plus 1 because we are multiplying the numerator

and the denominator by x plus 1. So distribute that three and what we’re

going to get is a 3x plus 3 in the numerator all over the denominator of x

plus 1 times x minus 6. And now, because we have a common denominator, I can just

continue this because we know when we add or subtract fractions you have a

common denominator and then you add or subtract straight across the numerator.

So I’m just going to continue this minus because of this minus sign. Now remember

we are subtracting that entire numerator so that’s why I put that in parenthesis – so minus the quantity x + 15. Here let’s think about this minus sign;. You’re going

to distribute that – you can give it as a negative one if you’d like

to, whatever helps you – but this is going to change the sign of everything inside

the parentheses so the result now is going to be 3x plus 3 minus x minus 15

all over that common denominator of x plus 1 times x minus 6. And by the way,

it’s okay to write x plus 1 before x minus 6. Just like 8 times 5 is equal to

5 times 8, this is the commutative property of multiplication – you can write

these in the opposite order. Now let’s combine some like terms. So 3x minus x

gives us 2x and positive 3 minus 15 gives you negative 12 all over that

common denominator of x plus 1 times x minus 6. Notice that i did not multiply

those factors in the denominator and, I’ll show you why right now, when you

simplify a rational expression you want to factor both the numerator and the

denominator. So that’s why we left that denominator factored because we were

going to end up simplifying that anyway. And now let’s factor the numerator and

that is 2, is the common factor that we can factor out, times x minus 6, is the

quantity that’s left. So 2 times the quantity x minus 6 equals 2x minus 12 over

that denominator x plus 1 times the quantity x minus 6. Hey surprise surprise –

we have a factor that is common that we can cancel out of the numerator and the

denominator, and look out for that – that’s probably going to happen a lot in

problems. They’re kind of set up so that you can cancel something out – not always,

but look for it. So we have 2 over x plus 1 as our final answer.

If you asked me if I knew how to do this, I would have said I'd forgotten this decades ago. But it turns out, with a little assist from the narrator, I actually remember. Cool.