Range of validity for Binomial Series

1. Oct 11, 2014

LiHJ

1. The problem statement, all variables and given/known data

Dear Mentors and Helpers,
here's the question:
Find the range of validity for (1 + 3x/2)^(-1) and (1 + 1/(3x))^(-1).
2. Relevant equations

3. The attempt at a solution
For the first binomial series:

-1 < 3x/2 < 1
-2 < 3x < 2 (multiply 2 throughout)
-2/3 < x < 2/3 (divide by 3 throughout)

For the second binomial series:
-1 < 1/(3x) < 1
-3x < 1 < 3x (multiply 3x throughout)

-3x < 1 or 1 < 3x
x > -1/3 (divide -3, inequality change sign) or 1/3 < x (divdide 3)

Therefore I get: x > -1/3 or x > 1/3

However my range isn't correct for the second binomial series can any Mentors or PF helper guide me and correct my mistakes.

Thank you

2. Oct 11, 2014

HallsofIvy

Staff Emeritus
For the second series you multiply each part of the inequality by 3x. When you multiply by -3, you note that it is negative so you change the direction of the inequality- but 3x might be negative also. So you must also consider the sign of 3x.

If 0< 1/3x< 1 then 3x is positive so 0< 1< 3x, x> 1/3. If -1< 1/3x< 0 then 3x is negative so -3x> 1> 0 and x<-1/3, not ">".

3. Oct 11, 2014

LiHJ

Thank you for the explanation, I finally understand :w