Step 1:

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.

Step 2:

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 $y \geq 2$

Step 3:

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)

Step 4:

Hence the solution of the given system of linear inequalities is represented by the common shaded region, including the points on the lines.