I'm trying to solve median residual life defined as
$$ \mathrm{mrl}(x) = \mathrm{median} ( X - x | X > x ) $$
where "solve" means to simplify with survival function $S(x) = P(X > x)$ or hazard rate $h(x) = \lim_{\Delta x \rightarrow 0} \frac{P(x \le X < x + \Delta x | X \ge x)}{\Delta x}$
What I tried:
Define $G(\xi ; x)$ be CDF of the random variable $X - x | X >x$, then $$G(\xi ; x) = \frac{1 - S(x + \xi)}{S(x)}$$
What I want is median, which I denote with $m$, then $$G(m ; x ) = \frac{1 - S(x + m)}{S(x)} \overset{!}{=} 0.5$$
Then I know given $x$, the $m = mrl(x)$, but I have difficulty dealing with those inverse function $S^{-1}(x)$.
Instead of defining $G(\xi; x)$, observe
$$ P[X-x > \xi | X > x ] = S(x+\xi)/S(x) $$
Note : $\mathrm{mrl}(x) = m(x)$ such that $S(x+m(x) )/S(x) = 0.5$
Thus,
$$ mrl(x) = m(x) =S^{-1}(S(x)/2) - x $$