Which is the solution region of (x+y\geq 4), (x+y<4), (x\geq 0)?

Correct answer: Empty set

Direct answer: Option B, the empty set. The first inequality says \(x+y\geq4\), so the sum must be 4 or more. The second says \(x+y<4\), so the same sum must be strictly less than 4. No number can be both at least 4 and less than 4. Therefore no point \((x,y)\), even one satisfying \(x\geq0\), can satisfy all the inequalities. Option B is correct. Option A, the line \(x+y=4\), is not included because the strict inequality \(x+y<4\) excludes the boundary, and the first inequality alone cannot rescue it. Option C, the first quadrant, contains many points with sums below 4 and many with sums above 4, but it does not satisfy both contradictory conditions. Option D, the region above the line, is also not a solution because one inequality requires one side of the line while the other requires the opposite side, with no overlap. The condition \(x\geq0\) becomes irrelevant after the contradiction is found. Memory cue: \(S\geq4\) and \(S<4\) together always give no solution.