For the inequality (x+2y\leq 8) with boundary (x+2y=8), which point is in the solution region?

Correct answer: ((0,0))

To identify the solution region of an inequality, substitute a convenient test point into the inequality. The point belongs to the region if the resulting statement is true. The boundary line is \(x+2y=8\), and the inequality includes all points for which \(x+2y\) is at most 8. Because the symbol is \(\leq\), points on the boundary are included as well.

Test the options. For \((0,0)\), we get \(0+2(0)=0\), and \(0\leq8\) is true. Thus option B lies in the solution region. The point \((5,3)\) gives 11, \((9,1)\) gives 11, and \((4,4)\) gives 12; all are greater than 8. Hence those points are outside the required region.