x/(x-3) subtracted from (x-2)/(x+3)?

rational expressions

2 Answers
Nov 17, 2017

-(8x-6)/((x+3)(x-3))

Explanation:

"before we can subtract the fractions we require "
"them to have a "color(blue)"common denominator"

"this can be achieved as follows"

"multiply numerator/denominator of "(x-2)/(x+3)" by "(x-3)

"multiply numerator/denominator of "x/(x-3)" by "(x+3)

rArr(x-2)/(x+3)-x/(x-3)

=((x-2)(x-3))/((x+3)(x-3))-(x(x+3))/((x+3)(x-3))

"now the denominators are common subtract the numerators"
"leaving the denominator as it is"

=(cancel(x^2)-5x+6cancel(-x^2)-3x)/((x+3)(x-3))

=(-8x+6)/((x+3)(x-3))=-(8x-6)/((x+3)(x-3))

"with restrictions on the denominator "x!=+-3

Nov 17, 2017

(-8x+6)/((x+3)(x-3))

Explanation:

In order to subtract fractions, we have to make sure the denominators (i.e, the bottom part of the fractions) are the same. We are given:

(x-2)/(x+3)-x/(x-3)

Notice that the denominators are different. The goal is to find the Least Common Multiple. A common denominator of both (x+3) and (x-3) is some value that has both those numbers as a multiple. The fastest, easiest number that is a multiple of both (x+3) and (x-3) is the value:

(x+3)(x-3)

Next, convert both fractions by multiplying (both numerator and denominator) by the missing multiple. Here is what that looks like:

(x-2)/(x+3)*color(red)(x-3)/color(red)(x-3)-(x)/(x-3)*color(red)(x+3)/color(red)(x+3)

Rewriting gives

((x-2)(x-3))/((x+3)(x-3))-(x(x+3))/((x+3)(x-3))

Now that the denominators are the same value, we can subtract them

((x-2)(x-3)-x(x+3))/((x+3)(x-3))

Simplifying the numerator requires using FOIL and the distributive law.

(x^2-3x-2x+6-x^2-3x)/((x+3)(x-3))

Combining like terms, we get

(-8x+6)/((x+3)(x-3))