The functional equation \(f(xy)+f(x+y)=f(xy+x)+f(y)\) was introduced by \textit{T. M. K. Davison} [Problem 191R1. Aequationes Math. 20, 306 (1980)]. Its general solution was recently given by \textit{R. Girgensohn} and \textit{K. Lajkó} [ibid. 60, No.~3, 219--224 (2000; Zbl 0970.39017)]. The stability of this equation was investigated by \textit{S.-M. Jung} and \textit{P. K. Sahoo} [J. Math. Anal. Appl. 238, No.~1, 297--304 (1999; Zbl 0933.39052); Kyungpook Math. J. 40, No.~1, 87--92 (2000; Zbl 0967.39010)]. Let \(F\) be a ring with the unit element, let~\(E\) be a~Banach space and let \(\varphi:F\times F\to[0,\infty)\) satisfies the condition \(\sum_{n=1}^{\infty}2^{-n}\varphi(2^{n-1}x,2^{n-1}y+z)<\infty\) for all \(x,y,z\in F\). The main result of the paper is the following {Theorem.} If a function \(f:F\to E\) satisfies the inequality \[ \bigl\| f(xy)+f(x+y)-f(xy+x)-f(y)\bigr\| \leq\varphi(x,y) \] for all \(x,y\in F\), then there exists a~unique additive function \(A:F\to E\) such that \[ \bigl\| f(6x)-A(x)-f(0)\bigr\| \leq\sum_{n=0}^{\infty}\frac{M(2^nx)}{2^n} \] for all \(x\in F\), where \[ \begin{multlined} M(x)=\frac{1}{2}\bigl[ \varphi(4x,-4x)+\varphi(4x,-4x+1)+\varphi(8x,-2x) +\varphi(3x,0)+\varphi(3x,1)\\ +\varphi(6x,0)+\varphi(7x,-x)+\varphi(7x,-4x+1) +\varphi(14x,2x) \bigr]. \end{multlined} \] The stability of the pexiderized version of Davison's equation \(f(xy)+g(x+y)=h(xy+x)+k(y)\) is also proved.