The vertices are found by replacing the inequalities with equalities.
-x+7=(x+3)/4, -4x+28=x+3, 5x=25, x=5, y=-5+7=2. Call this A(5,2).
-x+7=⅔x+2, 5x/3=5, x=3, y=-3+7=4. Call this B(3,4).
⅔x+2=(x+3)/4, 8x+24=3x+9, 5x=-15, x=-3, y=-2+2=0. Call this C(-3,0).
Although ABC is a triangle with the above coordinates for the vertices, all the inequalities are not satisfied by the area of the triangle because y>-x+7 is beyond the side BC of the triangle. If the inequality were to be reversed to y<-x+7, the internal area of the triangle (excluding the lines themselves) would satisfy all the inequalities.