Question

The graph of the inequation \( 2x + 3y>6 \) is the:

• A. Entire XOY-plane
• B. Half-plane that contains the origin
• C. Half-plane that neither contains the origin nor the points on the line \( 2x + 3y = 6 \)
• D. Whole XOY-plane excluding the points on the line \( 2x + 3y = 6 \)

Hint:

For linear inequalities, test a point (e.g., the origin) to determine which half-plane satisfies the inequality.

Answer 1

The Correct Option is C

Step 1: Rewrite the inequality: \[ 2x + 3y>6 \quad \Rightarrow \quad y>-\frac{2}{3}x + 2. \] Step 2: The inequality represents the region above the line \( 2x + 3y = 6 \), excluding the points on the line. 
Since \( 2(0) + 3(0) = 0 \not > 6 \), the origin is not part of the solution.