Linear Inequalities in Space

Which two of the three points $A(0,2,-1), B(0,-3,2), C(-3,2,1)$ lie on the same side of the plane $3 x+2 y+z+1 = 0$?

Which of the two planes $x+2 y+4 z-11 = 0$ and $x-y+3 z-9 = 0$ separates the points $A(1,3,-1)$ and $B(5,-3,1)$?

Two points $A(1,3,-1)$ and $B(5,3,1)$ and a plane $x-4 y + 2 z-7 = 0$ are given. Determine whether the given plane intersects the segment $A B$, its extension beyond the point $A$, or beyond the point $B$.

Two points $P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ and a plane $A x + B y+C z+D = 0$ are given. Without finding its point of intersection with the line $P Q$, find the ratio in which this point divides the segment $P Q$.

Two points $P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ are given. Find in what ratio the point of intersection of the line $P Q$ with the plane

divides the segment $P Q$.

Prove that the vector with coordinates $(A, B, C)$ is always directed toward the "positive" half-space relative to the plane $A x+B y+C z+D = 0$.

A point $\left(x_{0}, y_{0}, z_{0}\right)$ is given. Under what necessary and sufficient condition does this point lie between the two parallel planes $A x+B y+C z+D = 0$ and $A x+B y+C z+E = 0$?

Find the necessary and sufficient condition for a point to lie inside the tetrahedron formed by the planes

Four planes are given: $x+y = 0, x-z = 0, x+y+2 z = 0$, $y = 0$. In each of the following cases, find planes passing through the point $P$ such that the quadrilaterals cut out on them by these planes are parallelograms, and determine whether the point $P$ lies inside or outside these parallelograms: