Is it true that |r|=|s| (is equivalent to ) r=|s|?

Question

I'm actually doing an exercise where I have to draw graphs of functions. I understand r=|s| but not |r|=|s|. Are they the same?

Answer 1

They are not the same. If $r=|s|$, then $r$ can never be negative, but $|r|=|s|$ is true if $r=-1$ and $s=1$ (or for that matter if $s=-1$). The statement that $|r|=|s|$ just says that $r=\pm s$, i.e., that $r=s$ or $r=-s$: in both cases $r$ and $s$ will have the same absolute value, regardless of their algebraic signs.

Answer 2

Let $r = 4$ and $s = -4$. Then, $r = |s|$ and $|r| = |s|$.

Then, switch it around. Let $r = -4$ and $s = 4$. Then, $|r| = |s|$ but $r \neq |s|$.

So they are not always the same.