What is the common solution of (x+y\le 6) and (x+y\ge 6)?

Correct answer: line (x+y=6)

The direct answer is option B: the line \(x+y=6\). A point satisfies both inequalities only if it is at most 6 and at least 6 at the same time. Therefore its value must be exactly 6: \(x+y=6\). Geometrically, \(x+y\le6\) is the half-plane on one side of the line, while \(x+y\ge6\) is the half-plane on the other side. Their common part is only the boundary line, and both include that boundary because the signs contain equality. Option A is wrong because points with \(x+y<6\) fail the second inequality. Option B is correct because every point on the line satisfies both. Option C is wrong because many points do satisfy both, for example \((0,6)\). Option D is wrong because the origin has sum 0, not 6. Memory cue: a quantity that is both no greater than and no smaller than a number must equal that number.