SOLUTION: This is a problem I need help with.[(1/9)+(1/3x)] / [(x/9)-(1/x)]

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: This is a problem I need help with.[(1/9)+(1/3x)] / [(x/9)-(1/x)]      Log On


   

Question 186669: This is a problem I need help with.[(1/9)+(1/3x)] / [(x/9)-(1/x)]
Found 2 solutions by vleith, jim_thompson5910:
Answer by vleith(2983) About Me  (Show Source):
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
In order to simplify %281%2F9%2B1%2F%283x%29%29%2F%28x%2F9-1%2Fx%29 let's simplify the numerator and denominator separately first


Simplifying the numerator 1%2F9%2B1%2F%283x%29:



1%2F9%2B1%2F%283x%29 Start with the numerator. Take note that the LCD 9x%2F9x



x%2F%289x%29%2B1%2F%283x%29 Multiply the first fraction by x%2Fx


x%2F%289x%29%2B3%2F%289x%29 Multiply the second fraction by 3%2F3


%28x%2B3%29%2F%289x%29 Add the fractions.


So 1%2F9%2B1%2F%283x%29=%28x%2B3%29%2F%289x%29 where x%3C%3E0


----------------------------------------------------------


Simplifying the denominator x%2F9-1%2Fx:


x%2F9-1%2Fx Start with the denominator. Take note that the LCD 9x%2F9x


%28x%5E2%29%2F%289x%29-1%2Fx Multiply the first fraction by x%2Fx


%28x%5E2%29%2F%289x%29-9%2F%289x%29 Multiply the second fraction by 9%2F9


%28x%5E2-9%29%2F%289x%29 Subtract the fractions.


So x%2F9-1%2Fx=%28x%5E2-9%29%2F%289x%29 where x%3C%3E0



---------------------------------------------------------------


Now that we've simplified the numerator of %281%2F9%2B1%2F%283x%29%29%2F%28x%2F9-1%2Fx%29, we then get:


%28%28x%2B3%29%2F%289x%29%29%2F%28%28x%5E2-9%29%2F%289x%29%29




%28%28x%2B3%29%2F%289x%29%29%2F%28%28x%5E2-9%29%2F%289x%29%29 Start with the simplified expression.


%28%28x%2B3%29%2F%289x%29%29%2A%28%289x%29%2F%28x%5E2-9%29%29 Multiply the first fraction by the reciprocal of the second fraction.


%28%28x%2B3%29%2F%289x%29%29%2A%28%289x%29%2F%28%28x%2B3%29%28x-3%29%29%29 Factor x%5E2-9 to get %28x%2B3%29%28x-3%29


%289x%28x%2B3%29%29%2F%289x%28x%2B3%29%28x-3%29%29%29 Combine the fractions.


Highlight the common terms.


Cancel out the common terms.


1%2F%28x-3%29 Simplify



==================================================================

Answer:

So %281%2F9%2B1%2F%283x%29%29%2F%28x%2F9-1%2Fx%29 simplified to %28%28x%2B3%29%2F%289x%29%29%2F%28%28x%5E2-9%29%2F%289x%29%29 which in turn simplified to 1%2F%28x-3%29



So in short, %281%2F9%2B1%2F%283x%29%29%2F%28x%2F9-1%2Fx%29 simplifies to 1%2F%28x-3%29



In other words, %281%2F9%2B1%2F%283x%29%29%2F%28x%2F9-1%2Fx%29=1%2F%28x-3%29 where x%3C%3E-3, x%3C%3E0, or x%3C%3E3