Verifying that a point is satisfied by inequalities

Questions
1

Given a list of inequalities, I would like to check to see if a point is satisfied by said list of inequalities.

Example:
{x >= 0, y >= 0, z >= 0, 2 x >= y, 2 z >= y, 2 x + y <= 4,
2 z + y <= 4};
Check {x,y,z}./{2,4,0}. Which would return False.
Check {x,y,z}./{0,0,0}. Which woudl return True.

Is this possible?

2

ineq = {x >= 0, y >= 0, z >= 0, 2 x >= y, 2 z >= y, 2 x + y <= 4,
2 z + y <= 4};
And @@ (ineq /. Thread[{x, y, z} → {2, 4, 0}])
And @@ (ineq /. Thread[{x, y, z} → {0, 0, 0}])