We study the existence and uniqueness of weak and entropy solutions for the nonlinear inhomogeneous Neumann boundary value problem involving the 𝑝(𝑥)-Laplace of the form − div ɑ(𝑥, ∇𝑢) + |𝑢| 𝑝(𝑥)−2 𝑢 = f in Ω, ɑ(𝑥, ∇𝑢).η = 𝜑 on ∂Ω, where Ω is a smooth bounded open domain in ℝN, N ≥ 3, 𝑝 ∈ C(Ω) and 𝑝(𝑥) > 1 for 𝑥 ∈ Ω. We prove the existence and uniqueness of a weak solution for data 𝜑 ∈ L(𝑝−) ′ (∂Ω) and f ∈ L(𝑝−) ′ (Ω), the existence and uniqueness of an entropy solution for L1−data f and 𝜑 independent of 𝑢 and the existence of weak solutions for f dependent on 𝑢 and 𝜑 ∈ L(𝑝−) ′ (Ω).