about decimal places

Question

Suppose that $\tilde x$ is an approximation to x. Given that a necessary and sufficient condition for $n$ correct decimal places in $\tilde x$ is $|x − \tilde x| < 0.5 \times 10^{−n}$, so, how to derive a sufficient condition for $n$ correct significant figures in $\tilde x$, in which the upper bound on the relative error does not depend on $x$.

Answer 1

You can't. Relative error is defined as error divided by the actual value. If $x \approx 10^9$ you can get five significant figures with an absolute error of $5\cdot 10^4$. If $x \approx 10^{-9}$ to get five significant figures you need an absolute error of $5 \cdot 10^{-14}$