I cannot understand the point on p51 in the Bertsimas -book. You have a point $D=(0,0,0)$. Suppose you replace $x_1+x_2+x_3=1$ with $x_1+x_2+x_3\leq 1$ and $x_1+x_2+x_3\geq 1$. Now all of a sudden D transformed from non-basic solution into basic solution, why? $x_1+x_2+x_3\geq 1$ is not active with $(0,0,0)$ but the other is so the new two constraints are not active with logical AND but they are active with logical OR. But the LP -book uses the word AND instead of OR so dizzied!
Related book parts with pictures
CRUX-POINT: why this kind of distinction between the terms "basic" solution and "basic feasible" solution?
shoda explained that some Simplex methods exploits the basic -solutions in the intermediate steps. I think this is the crux point, it is the reason why we have this kind of odd distinction. Please, elaborate on this further -- which Simplex -methods have intermediates with basic solutions, not feasible basic solutions. Compare and contrast, thank you.