  This paper deals with the quasilinear attraction-repulsion chemotaxis system
\begin{align*} \begin{cases}
  u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u
  -\chi u(u+1)^{p-2}\nabla v
  +\xi u(u+1)^{q-2}\nabla w\big)
  +f(u),
  \\[1.05mm]
  0=\Delta v+\alpha u-\beta v,
  \\[1.05mm]
  0=\Delta w+\gamma u-\delta w \end{cases} \end{align*} in a bounded domain
$\Omega \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) with smooth boundary
$\partial\Omega$, where $m, p, q \in \mathbb{R}$, $\chi, \xi, \alpha, \beta,
\gamma, \delta>0$ are constants. Moreover, it is supposed that the function $f$
satisfies $f(u)\equiv0$ in the study of boundedness, whereas, when considering
blow-up, it is assumed that $m>0$ and $f$ is a function of logistic type such
as $f(u)=\lambda u-\mu u^{\kappa}$ with $\lambda \ge 0$, $\mu>0$ and $\kappa>1$
sufficiently close to~$1$, in the radially symmetric setting. In the case that
$\xi=0$ and $f(u) \equiv 0$, global existence and boundedness have been proved
under the condition $p<m+\frac2n$. Also, in the case that $m=1$, $p=q=2$ and
$f$ is a function of logistic type, finite-time blow-up has been established by
assuming $\chi\alpha-\xi\gamma>0$. This paper classifies boundedness and
blow-up into the cases $p<q$ and $p>q$ without any condition for the sign of
$\chi\alpha-\xi\gamma$ and the case $p=q$ with $\chi\alpha-\xi\gamma<0$ or
$\chi\alpha-\xi\gamma>0$.
