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