  We prove the existence of multiple solutions for the following sixth-order
$p(x)$-Kirchhoff-type problem: $-M(\int_\Omega \frac{1}{p(x)}|\nabla \Delta
u|^{p(x)}dx)\Delta^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u +
h(x) \ \ \mbox{on} \ \Omega$ and $ \ u=\Delta u=\Delta^2 u=0 \ \ \mbox{on} \
\partial\Omega,$ where $\Omega \subset \mathbb{R}^N$ is a smooth bounded
domain, $N > 3$, $\Delta_{p(x)}^3u = \operatorname{div}\Big(\Delta(|\nabla
\Delta u|^{p(x)-2}\nabla \Delta u)\Big)$ is the $p(x)$-triharmonic operator,
$p,q,r \in C(\overline\Omega)$, $1< p(x) < \frac N3$ for all $x\in
\overline\Omega$, $M(s) = a - bs^\gamma$, $a,b,\gamma>0$, $\lambda>0$, $g:
\Omega \times \mathbb{R} \to \mathbb{R}$ is a nonnegative continuous function
while $f,h : \Omega \times \mathbb{R} \to \mathbb{R}$ are sign-changing
continuous functions in $\Omega$. To the best of our knowledge, this paper is
one of the first contributions to the study of the sixth-order $p(x)$-Kirchhoff
type problems with sign changing Kirchhoff functions.
