We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.

Which positive real number x has the property that x, |x| , and x-|x| form a geometric progression (in that order)?

 Mar 15, 2018

x=0 is the only solution.


proof- |x|=q*x, x-|x|=q2*|x|.


if x=>0:

|x|=x. therefore x=x*q, x-|x|=x-x=0=q2*x. one solution to this equation is x=0. if x!=0 then we get q2=0 (by dividing the equation 0=x*q2 by x) meaning that q=0. but from that we can derive that x=0 (because of the first equation). that is a contradiction, meaning that if x=>0 then x=0.


if x<0:

|x|=-x meaning that q=-1. x-|x|=x-(-x)=2x=q2*|x|=(-1)2*(-x)=-x. this means that 3x=0 but then x=0 and that is a contradiction to x<0.


therefore, the only real number that satisfies the conditions of your question is x=0.

 Apr 17, 2018

17 Online Users