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

Question

Kennedy Adams · Accepted Answer

#(2x)/(x-6)#

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#.

Julia Adams · Answer

undefined To combine the expressions (x-2)/(x-6) and (x+2)/(6-x), we need to find a common denominator. In this case, the denominators are already the same, but with opposite signs.

To combine them, we can rewrite the second expression as -(x+2)/(x-6), since a negative sign can be factored out.

Now, we can add the two expressions together: (x-2)/(x-6) + (-(x+2)/(x-6)).

Combining the numerators, we get (x-2 - (x+2))/(x-6).

Simplifying the numerator, we have (x-2 - x - 2)/(x-6).

This further simplifies to (-4)/(x-6).

Therefore, the combined expression is (-4)/(x-6).