  In this article we find necessary and sufficient conditions for the strong
maximum principle and compact support principle for non-negative solutions to
the quasilinear elliptic inequalities $$\Delta_\infty u + G(|Du|) - f(u)\,\leq
0\quad \text{in}\; \mathcal{O},$$ and $$\Delta_\infty u + G(|Du|) - f(u)\,\geq
0\quad \text{in}\; \mathcal{O},$$ where $\mathcal{O}$ denotes the infinity
Laplacian, $G$ is an appropriate continuous function and $f$ is a
nondecreasing, continuous function with $f(0)=0$.
