Our main result concerns a perturbation of a classic theorem of Khintchine in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers (ψn) n∈N and differentiable functions (ϕn : J → R) n∈N so that for Lebesgue-a.e. θ ∈ J, the inequality nθ + ϕn(θ) ≤ ψn has infinitely many solutions. The main novelty is that the magnitude of the perturbation |ϕn(θ)| is allowed to exceed ψn, changing the usual "shrinking targets" problem into a "shifting targets" problem. As an application of the main result, we prove that if the linear equation y = ax + b, a, b ∈ R, has infinitely many solutions in N, then for Lebesgue-a.e. α > 1, it has infinitely many or finitely many solutions of the form ⌊n α ⌋ according as α < 2 or α > 2.