MATHEMATICS · Single correct · CBSE · Mathematics · Linear Programming · Linear Programming
Problem
Suppose you are graphing the inequality 2x−3y≤6. After drawing its boundary line 2x−3y=6, you choose the point (0,0) as a test point. Which of the following best explains why this choice helps determine the half-plane to shade?
(a) Because (0,0) always belongs to the solution set of any linear inequality, the half-plane containing it is shaded.
(b) Substituting (0,0) gives 2⋅0−3⋅0=0, the same as the right side if the constant is ignored, so the point is on the boundary and tells us whether to draw the line solid.
(c) Substituting (0,0) into 2x−3y≤6 shows whether the inequality is true there; this truth value identifies which half-plane contains all solutions.
(d) Because (0,0) makes the inequality 0≤6 true, we must shade the side of the line that does NOT contain the origin to avoid counting boundary points twice.