  We derive a family of weighted Hardy-type inequalities in the variable
exponent Lebesgue space with an additional term of the form
  \[ \int_\Omega\ |\xi|^{p(x)} \mu_{1,\beta}(dx)\leqslant \int_\Omega |\nabla
\xi|^{p(x)}\mu_{2,\beta}(dx)+\int_\Omega \left|\xi{\log \xi} \right|^{p(x)}
\mu_{3,\beta}(dx), \] where $\xi$ is any compactly supported Lipschitz
function. The involved measures depend on a certain solution to the partial
differential inequality involving $p(x)$-Laplacian ${-}\Delta_{p(x)} u\geqslant
\Phi$, where $\Phi$ is a given locally integrable function, and $u$ is defined
on an open and not necessarily bounded subset $\Omega\subseteq\mathbb{R}^n $,
and a certain parameter $\beta$. We derive new Caccioppoli-type inequality for
the solution $u$. As its consequence we get Hardy-type inequality. We
illustrate the result by several one-dimensional examples.
