Question

How do you subtract `\frac { 3} { x - 2} - \frac { x + 2} { x }`?

Answer 1

`3/(x-2) - (x+2)/x = -(x^2-3x-4)/(x^2-2x)`

Explanation:

A fraction construct is such that we have:

`("count")/("size indicator")->("numerator")/("denominator")`

You can not DIRECTLY add or subtract 'counts' unless the 'size indicators' are the same

`color(green)([3/(x-2)color(red)(xx1)] -[(x+2)/xcolor(red)(xx1)]`

`color(green)([3/(x-2)color(red)(xx x/x)] -[(x+2)/xcolor(red)(xx(x-2)/(x-2))]`

Note that `(x+2)(x-2) = x^2-2^2`

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

Now we can subtract the 'counts'

`(3x-x^2+4)/(x(x-2))`

`(-x^2+3x+4)/(x(x-2)`

`-(x^2-3x-4)/(x(x-2))`

Lets see if we can cancel anything through factoring

`-((x+1)(x-4))/(x(x-2))` Nothing immediately obvious

So `3/(x-2) - (x+2)/x = -(x^2-3x-4)/(x^2-2x)`