Absolute error bound for a perturbed linear equation

Absolute error bound for a perturbed linear equation

Consider a perturbed linear equation
\begin{equation} (A+\delta A)(x + \delta x)=(b+\delta b) \end{equation}
where $Ax=b$.
It is well-known that a relative error bound for the solution $x+\delta x$
can be given by
\begin{equation} \frac{\|\delta x\|}{\|x\|}\leq \|A\|\|A^{-1}\|
\left(\frac{\|\delta b\|}{\|b\|}+\frac{\|\delta A\|}{\|A\|}\right).
\end{equation}
However, in my case, the term $\|b\|$ can be very close to zero, and I
only care about the absolute error $\|\delta x\|$. Therefore, is there any
result to bound $\|\delta x\|$ which needs not to estimate $\|x\|$ and
$\|b\|$?