x/(x-3)xx−3 subtracted from (x-2)/(x+3)x−2x+3?
rational expressions
rational expressions
2 Answers
Explanation:
"before we can subtract the fractions we require "before we can subtract the fractions we require
"them to have a "color(blue)"common denominator"them to have a common denominator
"this can be achieved as follows"this can be achieved as follows
"multiply numerator/denominator of "(x-2)/(x+3)" by "(x-3)multiply numerator/denominator of x−2x+3 by (x−3)
"multiply numerator/denominator of "x/(x-3)" by "(x+3)multiply numerator/denominator of xx−3 by (x+3)
rArr(x-2)/(x+3)-x/(x-3)⇒x−2x+3−xx−3
=((x-2)(x-3))/((x+3)(x-3))-(x(x+3))/((x+3)(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"now the denominators are common subtract the numerators
"leaving the denominator as it is"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
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:
Notice that the denominators are different. The goal is to find the Least Common Multiple. A common denominator of both
Next, convert both fractions by multiplying (both numerator and denominator) by the missing multiple. Here is what that looks like:
Rewriting gives
Now that the denominators are the same value, we can subtract them
Simplifying the numerator requires using FOIL and the distributive law.
Combining like terms, we get