solving the inequality

Question

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$

Answer 1

Set $y = \dfrac{|x|-|1-x|}{|x|}= 1 - \lvert \frac1x-1 \rvert$.

Note $y > 1 $ is not possible, and $y \in [0, 1]$ when $x \ge \frac12$. Otherwise $y$ is negative, (so what?).

So the only case to consider is $y \in [0, 1]$, which means $2x-1< 8-x \implies ...$

Answer 2

One can distinguish multiple cases

• $x>0, \ 1-x>0$
• $x>0, \ 1-x<0$
• $x<0, \ 1-x>0$
• $x<0, \ 1-x<0$

As an example,when $x<0$ and $1-x<0$ we have that the inequality reads $\left(\frac{1-x-x}{-x}\right)^{2x-1}>\left(\frac{1-x-x}{-x}\right)^{8-x}$ or equivalently $\left(\frac{2x-1}{x}\right)^{2x-1}>\left(\frac{2x-1}{x}\right)^{8-x}$

It then follows that this is true for $2x-1>8-x$ or equivalently $3x>9 \rightarrow x>3$, but this is not possible since we found this solution under the assumption that $x<0$.

You could do the same for each case. This is the most intuitive solution, but Macavity's way is nicer.