I have the following inequality

$\displaystyle \frac{1}{2}e^{-|x|} > \frac{1}{4}e^{-\frac{|x-1|}{2}}$

By taking the log of both sides and some algebra, I get

$\displaystyle -2|x|-2log2 > -|x-1|-2log4$

At this point, I'm not sure what to do, since there's absolute values on both sides of the inequality, as well as other terms outside the absolute value. Any ideas? I know the approximate range is -2.4 < x < .8 . See the attached plot of the two functions.