sign function inequality $ -a (S-S*) sgn (S-S*)< -a(S-S*) ?$

Question

How it is possible that $-a (S-S*) x \leq -a (S-S*) sgn (S-S*)? $. Note that $x \leq1$ and $a>0$. Also $-a (S-S*) sgn (S-S*)\leq -a(S-S*)$ it is possible? Any other possible is there? $a$ and $S*$ are positive constants. $S$ and $x$ are variables in the domain, and satisfy the condition $0<S ≤ 1, 0<x ≤ 1.$