Final Answer:
(b) The solution lies in Quadrant II. because The solution lies in Quadrant II because it satisfies the conditions of both inequalities, with \(x\) negative and \(y\) positive.
Step-by-step explanation:
To solve the system of inequalities, we need to graph each inequality on the coordinate plane and find the overlapping region.
The first inequality, \(y > 2x + 4\), represents a half-plane above the line \(y = 2x + 4\). The second inequality, \(x + y < 6\), represents a half-plane below the line \(x + y = 6\).
By graphing these lines, we find that the overlapping region is in Quadrant II. The solution is above the line \(y = 2x + 4\) and below the line \(x + y = 6\). Therefore, the correct answer is (b) The solution lies in Quadrant II.
This conclusion can be reached by considering the signs of coefficients in the inequalities. In the first inequality, the coefficient of \(x\) is 2, indicating a positive slope. In the second inequality, the coefficients of \(x\) and \(y\) are both positive.
The common region satisfying both conditions is in Quadrant II, where \(x\) is negative, and \(y\) is positive. Thus, the final answer is (b) The solution lies in Quadrant II.
Therefore, the correct answer is option b.