Is {(x,y)∈R^2 | 2x+y≤2, x-y>4} an Open Subset of R^2?

[gavbacardi]

1. {(x,y)\in R^2 such that 2x+y<=2, x-y>4} Determine whether this subset of R^2 is open, closed or neither open nor closed.


2. I think this is an open subset but not sure how to prove it. I have rearranged the equations to give x>2, y<=-2x+1, y< x-1. I think it is open because x can get closer and closer to 2 but never equal it and y can get closer and closer to x-1 but never equal it. I'm not sure how prove this mathematically though?

Any help or hints would be great!

 

[STEMucator]

First things first. Draw what your set looks like by re-arranging your equations a bit. This will allow you to see whether or not your set is open.

Remember a set is open if every point is an interior point, that is for any point you choose, every neighborhood around it contains points only from the set.