Simple question about absolute value

Question

hi guys i was wondering about if

$|x| > z > 0 $

but

$|x - y | < | z - y|$

does it imply

$|x| < |z| => |x| < z$

and there a contradiction? :)

Answer 1

No. $|x-y|<|z-y|$ tells us that $y$ is closer to $x$ than it is to $z$. If $x>0$, this means simply that $y>\frac12(x+z)$, since $\frac12(x+z)$ is halfway between $x$ and $z$. Otherwise, $x<-z<0$, and it tells us similarly that $y<\frac12(x-z)$.

Answer 2

If $x=-2$, $y=-3$, $z=1$, then $$\begin{cases}0<1<\lvert -2\rvert\\ \lvert -2-(-3)\rvert<\lvert 1-(-3)\rvert\end{cases}$$