Graph linear inequalities and systems of linear inequalities as half-plane solution regions.
Graph a standard-form linear inequality
Problem
For \(2x+y\le~6\), identify slope-intercept boundary, \(x\)-intercept, \(y\)-intercept, style, and shading.
Big Picture
What this problem is really about
Standard form still describes one boundary line and one solution half-plane. We’ll replace the inequality sign by equality, rearrange carefully to expose the line’s signed slope, and find both axis intercepts as a geometric check. The inclusive symbol determines the boundary style, while a test point determines which side satisfies the original inequality and prevents a sign-based shading guess.
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