How is this absolute value inequality justified?

Question

In Spivak's Calculus, one of the proofs in chapter 5 says that $|y_{0}| - |y| \leq |y - y_{0}| < \frac{|y_{0}|}{2}$ implies that $|y| > |y_{0}|/2$. It might be obvious or something but I can't figure out how that leap is made.

Answer 1

You have $|y_0| - |y| < \frac{|y_0|}{2}$, so subtract $|y_0|$ from both sides and $-|y| < -\frac{|y_0|}{2}$. Then multiply through by $-1$ and don't forget to reverse the sense of the inequality.