Consider the inequality $3x +2y \leq 12$----(1)
Consider the corresponding equation $3x +2y =12$
The points $(4,0)$ and $(0,6)$ satisfy the equation.
The Graph of this line is shown in the diagram below.
The second inequality is $ x \geq 1$ ----(2)
Consider $x=1$ The graph of this line is shown in the diagram.
The third inequality is $y \geq 2$
consider $y=2$ The graph if this line is shown in the diagram
The inequality (1) represented the half plane below the line $3x+2y =12$ (including the line)
The inequality (2) represented the half plane on the right of the line $x=1$ (including the line x=1)
The inequality (3) represented the half plane below the line $y =2$ (including the line y=2)
Hence the solution of the given system of linear inequalities is represented by the common shaded region, including the points on the lines.