For the inequality (-x+y\le 1), what does the test point ((0,0)) indicate?

Correct answer: The side containing origin is correct

The direct answer is option A: the side containing the origin is correct. The boundary of \(-x+y\le1\) is the line \(-x+y=1\). To decide which side to shade, test the origin \((0,0)\). Substitution gives \(-(0)+0=0\), and \(0\le1\) is true. Therefore the origin satisfies the inequality, so the half-plane containing the origin is the solution side. Option A is correct. Option B is wrong because it says the origin side is false, whereas the test calculation proves it true. Option C is wrong because the inequality includes an entire half-plane, not only its boundary; the boundary alone would represent an equality. Option D is wrong because many points satisfy the inequality, for example the origin itself. When the test point lies on the boundary, choose another point, but here the origin gives a clear result. Memory cue: substitute the test point and shade the side that makes the inequality true.