Question

How do you solve `3/(x-2) = 1/(x-1) + 7/(x^2-3x+2)`?

Answer 1

`x=4`

Explanation:

Factorise the denominator of the last term by inspection:

`3/(x-2)=1/(x-1)+7/((x-2)(x-1))`

We need to get the `x`'s off the denominator to solve. I see lots of common terms, so to get rid of them, multiply both sides of the equation by `(x-2)(x-1)`:

`(3(x-2)(x-1))/(x-2)=(1(x-2)(x-1))/(x-1)+(7(x-2)(x-1))/((x-2)(x-1))`

Now we can cancel out a whole bunch of terms:

`(3cancel(x-2)(x-1))/cancel(x-2)=(1(x-2)cancel(x-1))/cancel(x-1)+(7cancel(x-2)cancel(x-1))/(cancel(x-2)cancel(x-1))`

And we are left with this:

`3(x-1)=(x-2)+7`

Expand the brackets, and put all the `x` terms on the left, and the constants on the right:

`3x-3=x+5`

`2x=8`