To determine whether the system of inequalities has solutions, let's analyze the inequalities:
- \( y > x + 2 \): This represents a region above the line \( y = x + 2 \).
- \( y < x - 2 \): This represents a region below the line \( y = x - 2 \).
Next, let's find the points where these two lines intersect to see if there's an overlap in the regions defined by the inequalities.
Finding the intersection points:
Set the two equations equal to each other: \[ x + 2 = x - 2 \]
Subtract \( x \) from both sides: \[ 2 = -2 \]
This is a contradiction, meaning that the lines \( y = x + 2 \) and \( y = x - 2 \) are parallel and will never intersect.
Analyzing the regions:
- The area described by \( y > x + 2 \) is above the line \( y = x + 2 \).
- The area described by \( y < x - 2 \) is below the line \( y = x - 2 \).
Since one region is above and the other is below, there cannot be any points that satisfy both inequalities simultaneously.
Conclusion: Yes, this system of inequalities has no solutions, as the two regions do not overlap at all.