To determine which option is a system of inequalities that has no solutions, we can analyze each option based on the relationships between the inequalities.
Option #1
- \( y > 23x + 3 \)
- \( y > 23x + 5 \)
For this option, both inequalities can be satisfied because the line \( y = 23x + 5 \) is above the line \( y = 23x + 3 \). Thus, there are solutions.
Option #2
- \( y > 23x + 3 \)
- \( y < 23x + 1 \)
In this case, the first inequality says \( y \) must be greater than a line with a slope of 23 and a y-intercept of 3, while the second inequality says \( y \) must be less than a line with the same slope and a y-intercept of 1. It is impossible for \( y \) to be greater than \( 23x + 3 \) and at the same time less than \( 23x + 1 \). Therefore, this system has no solutions.
Option #3
- \( y > 23x + 3 \)
- \( y < 23x + 7 \)
Here, the first inequality requires \( y \) to be above the line \( y = 23x + 3 \), and the second inequality requires \( y \) to be below the line \( y = 23x + 7 \). Since these two lines are parallel and \( 23x + 3 \) is below \( 23x + 7 \), there are values of \( y \) that can satisfy both inequalities. Therefore, this system has solutions.
Conclusion
The system of inequalities without solutions is:
The system of inequalities in Option #2 has no solutions.