Which point lies in the solution region of (x+2y\geq 4)?

Correct answer: ((0,3))

The direct answer is option D: \((0,3)\). To test a point in the solution region of \(x+2y\geq4\), substitute its x- and y-coordinates and check whether the inequality is true. For option A, \((0,0)\): \(0+2(0)=0\), and \(0\geq4\) is false. For option B, \((1,1)\): \(1+2(1)=3\), and \(3\geq4\) is false. For option C, \((2,0)\): \(2+2(0)=2\), and \(2\geq4\) is false. For option D, \((0,3)\): \(0+2(3)=6\), and \(6\geq4\) is true. Therefore D is the only point in the solution region. The boundary line is \(x+2y=4\), and because the sign is \(\geq\), the boundary is included and the side giving values at least 4 is shaded. Option A is wrong because its left side is 0, option B is wrong because it gives 3, and option C is wrong because it gives 2; all are less than 4. Option D works because its value is 6, which is greater than 4. A common mistake is to look only at positive coordinates; positive coordinates do not guarantee a solution. Always substitute. Memory cue: for a point question, calculate the left side first and compare it directly with the right side.