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Hello, I am stuck on something so simple. The problem is i have a great difficulty with the geometric interpretation of things.

So if I have a system of linear equations in three unknowns, like for example this:

-x - 2y + z = 0

x - 3y - 2z = 0

this is just a simple system of homogeneous equations. I can use simple Gaussian elimination to solve it. There's an infinite number of solutions to this and I can see the geometric reason: both of these equations represent a plane in space through the origin. The intersection of these two planes is a line, and there are an infinite number of points on that line.

I am starting to have difficulty when I think of solving this:

-x - 2y + z > 0

x - 3y - 2z > 0

Here it's the same equations only inequalities. I don't know how to envision this. When I think of the first inequality for example, I think this will be all the points (x,y,z) in the plane (-x -2y + z=0) that are above the line (2y + x). But I'm not sure. How do I represent a solution that satisfies these inequalities? I imagine I first solve the associated homogeneous equations that are above. Then where do I go from there?

So if I have a system of linear equations in three unknowns, like for example this:

-x - 2y + z = 0

x - 3y - 2z = 0

this is just a simple system of homogeneous equations. I can use simple Gaussian elimination to solve it. There's an infinite number of solutions to this and I can see the geometric reason: both of these equations represent a plane in space through the origin. The intersection of these two planes is a line, and there are an infinite number of points on that line.

I am starting to have difficulty when I think of solving this:

-x - 2y + z > 0

x - 3y - 2z > 0

Here it's the same equations only inequalities. I don't know how to envision this. When I think of the first inequality for example, I think this will be all the points (x,y,z) in the plane (-x -2y + z=0) that are above the line (2y + x). But I'm not sure. How do I represent a solution that satisfies these inequalities? I imagine I first solve the associated homogeneous equations that are above. Then where do I go from there?

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