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Let $\Gamma\subset \overline{\mathbb Q}^{\times}$ be a finitely generatedmultiplicative group of algebraic numbers. Let $\delta,\beta\in\overline{\mathbb Q}^\times$ be algebraic numbers with $\beta$irrational. In this paper, we prove that there exist only finitely many triples$(u, q, p)\in\Gamma\times\mathbb{Z}^2$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$such that $$ 0<|\delta qu+\beta-p|<\frac{1}{H^\varepsilon(u)q^{d+\varepsilon}},$$ where $H(u)$ denotes the absolute Weil height. As an application of thisresult, we also prove a transcendence result, which states as follows: Let$\alpha>1$ be a real number. Let $\beta$ be an algebraic irrational and$\lambda$ be a non-zero real algebraic number. For a given real number$\varepsilon >0$, if there are infinitely many natural numbers $n$ for which$||\lambda\alpha^n+\beta|| < 2^{- \varepsilon n}$ holds true, then $\alpha$ istranscendental, where $||x||$ denotes the distance from its nearest integer.When $\alpha$ and $\beta$ both are algebraic satisfying same conditions, then aparticular result of Kulkarni, Mavraki and Nguyen, proved in [3] asserts that$\alpha^d$ is a Pisot number. When $\beta $ is algebraic irrational, our resultimplies that no algebraic number $\alpha$ satisfies the inequality forinfinitely many natural numbers $n$. Also, our result strengthens a result ofWagner and Ziegler [6]. The proof of our results uses the Subspace Theorembased on the idea of Corvaja and Zannier [2] together with various modificationplay a crucial role in the proof.

On inhomogeneous extension of Thue-Roth's type inequality with moving targets