  For any linear inequality in three variables $\mathcal{L}$, we determine (if
it exist) the smallest integer $R(\mathcal{L}, \mathbb{Z}/3\mathbb{Z})$ such
that: for every mapping $\chi :[1,n] \to \{0,1,2\}$, with $n\geq R(\mathcal{L},
\mathbb{Z}/3\mathbb{Z})$, there is a solution $(x_1,x_2,x_3)\in [1,n]^3$ of
$\mathcal{L}$ with $\chi(x_1)+\chi(x_2)+\chi(x_3)\equiv 0$ (mod $3$). Moreover,
we prove that $R(\mathcal{L}, \mathbb{Z}/3\mathbb{Z})=R(\mathcal{L}, 2)$, where
$R(\mathcal{L}, 2)$ denotes the classical $2$-color Rado number, that is, the
smallest integer (provided it exist) such that for every $2$-coloring of
$[1,n]$, with $n\geq R(\mathcal{L}, 2)$, there exist a monochromatic solution
of $\mathcal{L}$. Thus, we get an Erd\H{o}s-Ginzburg-Ziv type generalization
for all lineal inequalities in three variables having a solution in the
positive integers. We also show a number of families of linear equations in
three variables $\mathcal{L}$ such that they do not admit such
Erd\H{o}s-Ginzburg-Ziv type generalization, named $R(\mathcal{L},
\mathbb{Z}/3\mathbb{Z})\neq R(\mathcal{L}, 2)$. At the end of this paper some
questions are proposed.
