Step 1:

The first inequality is $2x +y \geq 6$ -----(1)

Consider the equation $2x+ y =6$

The point $(3,0) $ and $(0,6)$ satisfy the equation.

The graph of this is given below, Consider the point (0,0)

we see that, $2(0) +0 \geq 6$

$0 \geq 6$ is false

The inequality $2x+y \geq 6$ represents the half plane not containing (0,0) .

It represents the region above the line $2x+y=6$ including the point on the line.

Step 2:

The second inequality is $3x+4y \leq 12$

the points (4,0) and (0,3) satisfy the equation.

The graph of the line $3x+4y =12$ is given below,

Consider the point (0,0)

we see that, $3(0)+4(0) \leq 12$

$ 0 \leq 12$ is true.

The inequality $3x+4y \leq 12$ represents the region below the line $3x+4y =12$ (including the points on the line) Containing the point (0,0)

Step 3:

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