Using the origin test relative to the line (2x-3y=12), which conclusion is correct for (2x-3y<12)?

Correct answer: The side containing the origin is shaded

For a linear inequality in two variables, the equation 2x-3y=12 first gives the boundary line. To identify the required half-plane, test a convenient point not on that line, usually the origin. Substitution gives 2(0)-3(0)=0, and 0<12 is true. Therefore the origin satisfies the inequality, so the half-plane containing the origin is the solution region. Because the sign is strict (<), the boundary line itself is not included and should normally be drawn dashed. Option B is correct. Option A reverses the test result, option C incorrectly includes only the line, and option D is false because many points, including (0,0), satisfy the inequality.