Range of validity for Binomial Series

[LiHJ]

Homework Statement

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

Homework Equations

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

 

[HallsofIvy]

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 ">".

 

[LiHJ]

Thank you for the explanation, I finally understand :w