Necessary conditions for solving equations including addition of multiple absolute value functions

Question

I am in a trouble with finding a necessary condition of have only one unique solution a equation: $$x -a= -\sum_{i}^{n}k_{i}\lvert b_{i}x-c_{i}\rvert$$ where $k_{i}\in(0,1)$. $$\\$$ I have tried Matlab simulation for some few values, and it looks a messy because the right hand side will be bent too many times. It seems that if we have: $$\sum_{i}^{n}\lvert k_{i}b_{i}\rvert<1$$ and the put the max point of the right hand side to the right of the $f(x)=x-a$, then it will give only one solution to the question but I am not quite sure. Also I want some serious mathematical proof$$\\$$ Does anyone have any idea about this? $$\\$$ Thanks for your help.