1. The problem asks which system of inequalities has no solution set. 2. A system of inequalities has no solution if the inequalities contradict each other, meaning there is no overlap in their solution regions. 3. Let's analyze each system: - System 1: $$2x + y > -5$$ $$2x + y \leq 3$$ These inequalities describe regions where $$2x + y$$ is greater than $$-5$$ and at the same time less than or equal to $$3$$. Since $$-5 < 3$$, there is an overlap, so this system has solutions. - System 2: $$2x + y \leq -5$$ $$2x + y < 3$$ Here, $$2x + y$$ must be less than or equal to $$-5$$ and also less than $$3$$. Since $$-5 < 3$$, the overlap is all values less than or equal to $$-5$$, so solutions exist. - System 3: $$2x + y \geq -5$$ $$2x + y > 3$$ Here, $$2x + y$$ must be greater than or equal to $$-5$$ and also greater than $$3$$. Since $$3 > -5$$, the overlap is all values greater than $$3$$, so solutions exist. - System 4: $$2x + y \leq -5$$ $$2x + y > 3$$ Here, $$2x + y$$ must be less than or equal to $$-5$$ and at the same time greater than $$3$$. These two conditions cannot be true simultaneously because $$-5 < 3$$. Therefore, there is no overlap, and this system has no solution. 4. Conclusion: The system with no solution set is: $$\begin{cases} 2x + y \leq -5 \\ 2x + y > 3 \end{cases}$$ This is because the inequalities contradict each other and cannot be satisfied at the same time.