00:01
Okay, the first step here is to rewrite the denominator as a linear factor.
00:10
So, in other words, we're going to factor the denominator.
00:14
So, there's many methods of factoring.
00:18
I use what's called the guess -and -check form.
00:24
You might have learned it a different way.
00:27
So, i know that both of these will be minus minus, because that's the only way to get a plus here.
00:35
And the only thing that multiplies to equal 3 is 1 and 3.
00:41
So, now the factors of 8 are 1 and 8, 2 and 4.
00:46
I just have to figure out which is going to give me an answer of 10 when i combine the inside and the outside.
00:53
So, i usually start with the smaller numbers.
00:57
So, i'll try 2 and 4, and then i'm just going to test the inside and the outside.
01:06
So, the inside gives me negative 4x, the outside gives me negative 6x, and negative 4x plus negative 6x is negative 10x.
01:18
So, i've found the correct factors.
01:26
Okay, so i know that i want to write this as two separate fractions, and each linear factor will be the denominator of one of those fractions.
01:44
And i know that this is going to be some constant a, and this is going to be some constant b.
01:50
I just have to figure out what those numbers are.
01:53
So, what i'm going to do is i'm going to multiply each part of this by 2x minus 1, 4x minus 3.
02:07
I'm going to do it to all three parts.
02:21
So, on the first fraction, the entire denominator cancels, and i'm left with just 66x.
02:30
On the second part of this, the 2x minus 1's cancel, and i'm left with a over a times 4x minus 3.
02:50
And on the last part, the 4x minus 3's cancel, and i'm left with b times 2x minus 1.
03:02
So, one way to figure out what x and what a and b are is to just try different numbers for x.
03:13
So, let's say to start with that x equals positive 3 4ths, and you'll see why i chose 3 4ths in just a second.
03:26
So, if i do 66 times 3 4ths equals a times 4 times 3 4ths minus 3 plus b times 2 times 3 4ths minus 1, watch what happens.
03:53
I get 99 halves.
03:58
Remember, you can cross simplify fractions.
04:01
This becomes 33, and this becomes 2 equals, well, 4 times 3 4ths is 3.
04:15
3 minus 3 is 0, and 0 times a is just going to be 0.
04:25
Two -thirds times 3 4ths is, again, let's cross cancel, three halves minus 1...