Question

The solution set of the inequality \(2x + 3y < 4\) is:

• A. an open half-plane not containing the origin.
• B. an open half plane containing the origin.
• C. a closed half plane containing the origin.
• D. the whole xy plane not containing the line 2x+3y=4.

Answer 1

The Correct Option is B

Given Inequality: \( 2x + 3y < 4 \)

Step 1: Identify the Boundary Line

The inequality is based on the line \( 2x + 3y = 4 \), which serves as the boundary dividing the plane into two regions.

Step 2: Check if the Line is Included

Since the inequality is "<" (not "≤"), the line itself is not included in the solution set. This means the boundary is not solid, and the region is open.

Step 3: Use a Test Point

Choose a simple point to test which side of the boundary is included. Let’s use the origin \( (0, 0) \):

• \( 2(0) + 3(0) = 0 \)
• \( 0 < 4 \) is true.

So the region includes the origin.

Step 4: Describe the Solution Set

The inequality \( 2x + 3y < 4 \) represents the region below the line \( 2x + 3y = 4 \), excluding the line itself. Since the origin is included, we conclude:

Final Answer: The solution set is an open half-plane containing the origin.