How to expand this expression: $\frac{z}{2}<|y|<z$, is it correct to expand it as following: $-z<-\frac{z}{2}<y<\frac{z}{2}<z$

Question

How to express this expression $\frac{z}{2}<|y|<z$.

Is it correct to expand it as following

$-z<-\frac{z}{2}<y<\frac{z}{2}<z$

Answer 1

Notice that $z > \vert y \vert \geq 0$. So we must have $z>0$ and $\vert y \vert > 0$. Note that $y = 0$ would imply $z$ is both positive and negative ... oops! For $y>0$, you have $$\frac{z}{2} < y < z$$ and for $y < 0$ you have $$ \frac{z}{2} < -y < z \implies \frac{-z}{2} > y > -z.$$ This shows that $y$ lies in $$\left(-z, \frac{-z}{2}\right) \cup \left(\frac{z}{2}, z \right)$$ So you need two statements to expand, and you would say $$-z<y<-\frac{z}{2}<0 \, \text{ or } \, 0<\frac{z}{2} < y < z.$$