Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
Find the overlap of three inequalities
Problem
For \(x\ge~0\), \(y\ge~0\), \(x+y\le~6\), identify empty/nonempty, boundedness, boundary inclusion, and vertices.
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What this problem is really about
The overlap must satisfy all three half-plane conditions simultaneously. We’ll use the two coordinate restrictions to locate the allowed quadrant, graph the remaining boundary from its intercepts, and keep every boundary allowed by an equality bar. Then the pairwise boundary intersections reveal the corners, while the enclosed directions determine whether the common region is empty, bounded, or unbounded.
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