Question

How do you combine `(x-2)/(x-6)-(x+2)/(6-x)`?

Answer 1

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

Explanation:

The trick here is to realize that you can write `6-x` as

`6 -x = -1 * (-6 + x) = -1 * (x-6)`

This means that the expression can be rewritten as

`(x-2)/(x-6) - (x+2)/(-1 * (x-6)) = (x-2)/(x-6) - (x+2)/(-(x-6))`

This will be equivalent to

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

You now have two fractions that have the same denominator, which means that you can go ahead and combine their numerators

`(x-2)/(x-6) + (x+2)/(x-6) = (x - color(red)(cancel(color(black)(2))) + x + color(red)(cancel(color(black)(2))))/(x-6) = color(green)(|bar(ul(color(white)(a/a)color(black)((2x)/(x-6))color(white)(a/a)|)))`

ALTERNATIVELY

You can also combine these two fractions by finding their common denominator, which in this case would be `(x-6)(6-x)`.

Multiply the first fraction by `1 = (6-x)/(6-x)` and the second fraction by `1=(x-6)/(x-6)` to get

`(x-2)/(x-6) - (x+2)/(6-x) = (x-2)/(x-6) * (6-x)/(6-x) - (x+2)/(6-x) * (6-x)/(6-x)`

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

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

Focus on the numerator first

`6x - x^2 - color(red)(cancel(color(black)(12))) + color(red)(cancel(color(black)(2x))) - x^2 + 6x - color(red)(cancel(color(black)(2x))) + color(red)(cancel(color(black)(12))) = -2x^2 + 12x`

This can be rewritten as

`-2x^2 + 12x = 12x - 2x^2 = 2x * (6 - x)`

The expression will once again be equal to

`(2x * color(red)(cancel(color(black)((6-x)))))/((x-6)color(red)(cancel(color(black)((6-x))))) = color(green)(|bar(ul(color(white)(a/a)color(black)((2x)/(x-6))color(white)(a/a)|)))`

Keep in mind that you need to have `x!=6`.