Question

The solution region of the inequality \( 2x + 4y \leq 9 \) is:

• A. open half plane containing origin
• B. closed half plane containing origin
• C. open half plane not containing origin
• D. closed half plane not containing origin

Answer 1

The Correct Option is B

To determine the solution region of the inequality \(2x + 4y \leq 9\), follow these steps: 

1. First, identify the boundary line associated with the inequality, which is \(2x + 4y = 9\).
2. Simplify the equation by dividing every term by 2: \(x + 2y = 4.5\).
3. Plot the line \(x + 2y = 4.5\) on the coordinate plane.
4. Determine if the inequality is less than or equal (\(\leq\)) by substituting the origin \((0,0)\) into the inequality:

\(2(0) + 4(0) \leq 9\Rightarrow 0 \leq 9\) (True)

1. Since the inequality holds true for the origin, the region containing the origin is part of the solution.
2. Since the inequality is \(\leq\), it includes the boundary line, indicating a closed half-plane.

Thus, the solution region is a closed half-plane containing the origin.

Answer 2

Rewrite the inequality as:

\[ 2x + 4y = 9 \quad \text{(boundary line)}. \]

Test the origin \((0, 0)\) in the inequality:

\[ 2(0) + 4(0) \leq 9 \quad \text{True.} \]

Thus, the solution includes the origin. Since the inequality is \(\leq\), the boundary line is included, and the solution is the closed half-plane containing the origin.