Solve inequality logarithm

I have a question about solving inequality logarithm with absolute x in it (attached in image). And i want to know if my work is right or not. Thanks. • Please have a look at the introduction to posting mathematical expressions. You will find there links to more advanced topics. Apr 8 '17 at 6:02
• I think if you're asking for help it is respectful to put your work in a legible format. It's expected here that you will learn Latex and type maths in that. Apr 8 '17 at 7:14
• Sorry. I can't put mathematic symbol here. i don't know how. Apr 8 '17 at 7:30

Assuming you want to show for what $x, \ log_{(1-|x|)}|(3x-1)|<1$ (with base 1-|x|), we have the following:

What is the inverse of $log|x|$? (i.e. its exponential form)

We also assume $logx$ has the base $e$ (when the base is not specified)

So upon inverting the logarithm to it's exponential equivalent, we have:

$$log_{(1-|x|)}|(3x-1)|<1\Rightarrow (1-|x|)<(3x-1)$$

Noting $logx$ is only defined for $x>0$, we thus immediately see x is bounded above by $1$. That is:

$$log(1-|x|) \Rightarrow x<1$$ Why? Consider $log(1-x)>0$

A lower bound for x follows from Considering the above inequality $(1-|x|)<(3x-1)$

Thus we have:

For $x>0$, $(1-x)<(3x-1)\Rightarrow 2<4x\Rightarrow x>\frac 12$

Thus the values for $x$, such that$\ log_{(1-|x|)}|(3x-1)|<1$ are $\frac 12<x<1$

• Apologies for the original solution, I think I am going crazy haha. Cheers! Apr 8 '17 at 6:03
• Wow! Thanks a lot. You really helpful. Could you tell me which part i got wrong in my work? Apr 8 '17 at 7:01
• You said the opposite, that is $(1-|x|)>(3x-1)$. Instead, first inverse the original logarithm into its exponential equivalent. (Take the logarithm's base to the power of the RHS (1), and set it equal to $3x-1$). Don't forget to accept and up vote this answer. Hope this help! Apr 8 '17 at 7:12
• Done! Thanks for the answer. Apr 8 '17 at 7:31