Question

How do you simplify `\frac { \frac { 6} { x - 5} + x } { \frac { 3} { x - 5} + 1}`?

Answer 1

`(6/(x-5)+x)/(3/(x-5)+1) = color(blue)(x-3`

Explanation:

First, let's move the denominator into a common denominator (`x-5`):

`(6/(x-5)+x)/((3+x-5)/(x-5))`

`(6/(x-5)+x)/((x-2)/(x-5))`

Now get the numerator into a common denominator (also `x-5`):

`((6+x^2-5x)/(x-5))/((x-2)/(x-5))`

We can now eliminate both denominators:

`((6+x^2-5x)/(cancel(x-5)))/((x-2)/(cancel(x-5)))`

`(x^2-5x+6)/(x-2)`

Factoring the numerator:

`((x-3)(cancel(x-2)))/(cancel(x-2))`

`color(blue)(ul(x-3`