To find the solution region for the inequality \(2x + 4y \leq 9\), perform the following steps:
- Identify the boundary line: \(2x + 4y = 9\).
- Simplify the boundary line equation by dividing by 2: \(x + 2y = 4.5\).
- Graph the line \(x + 2y = 4.5\) on the coordinate plane.
- Test the origin \((0,0)\) in the original inequality to determine which side of the line represents the solution:
\(2(0) + 4(0) \leq 9\)
\(0 \leq 9\) (This statement is True)
- Because the origin satisfies the inequality, the solution region includes the half-plane containing the origin.
- The "less than or equal to" symbol (\(\leq\)) indicates that the boundary line itself is part of the solution.
Therefore, the solution region is a closed half-plane that includes the origin.