What does the origin test give for \(x-2y\ge 6\)?

Correct answer: false

For the origin test, put \(x=0\) and \(y=0\). The inequality \(x-2y\ge 6\) becomes \(0\ge 6\), which is false. Therefore, the origin does not lie in the solution half-plane, so the region on the side of the line \(x-2y=6\) opposite to the origin is selected. Option A uses the same inequality but incorrectly calls it true. Exam tip: after substituting a test point, always check whether the resulting inequality is true or false.