Certainly! Let's walk through the solution to this question step by step.
### Understanding the Problem
The problem asks us to determine the condition under which a point is a solution to a system of two inequalities, based on the shaded regions created by these inequalities on a graph.
### Analysis of Possible Cases
When graphing two inequalities on the same coordinate plane, several scenarios can arise regarding where a point lies relative to these shaded regions:
1. A point lies in the shaded region of the top inequality, but not in the shaded region of the bottom inequality.
2. A point lies in the shaded region of the bottom inequality, but not in the shaded region of the top inequality.
3. A point lies in the shaded regions of both the top and bottom inequalities.
4. A point does not lie in the shaded region of the top or bottom inequalities.
### Determining the Solution
For a point to be a solution to the system of two inequalities, it must satisfy both inequalities simultaneously. Graphically, this means the point must lie in the overlap of the shaded regions of both inequalities.
### Conclusion
Based on our analysis, a point is a solution to a system of two inequalities if and only if it lies in the shaded regions of both the top and bottom inequalities. Therefore, the correct choice from the options provided is:
3. A point lies in the shaded regions of both the top and bottom inequalities.
This is the detailed, step-by-step reasoning to determine how we find a point that is a solution to the system of two inequalities.
