To determine which option is a system of inequalities that has no solutions, we need to look at the relationships between the lines represented by the inequalities.
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Option #1: \( y > \frac{2}{3}x + 3 \) and \( y > \frac{2}{3}x + 5 \)
Both inequalities represent lines with the same slope (\(\frac{2}{3}\)), but different y-intercepts. The second line \(y = \frac{2}{3}x + 5\) is above the first line \(y = \frac{2}{3}x + 3\). In this case, the region above both lines cannot exist simultaneously since any point satisfying the second inequality will not satisfy the first inequality. This option has no solutions.
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Option #2: \( y > \frac{2}{3}x + 3 \) and \( y < \frac{2}{3}x + 1 \)
Here, one line has a greater y-intercept (3) than the other (1). The first line is positioned above the second line. Therefore, there is no point that can satisfy being both above the first line and below the second line. This option also has no solutions.
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Option #3: \( y > \frac{2}{3}x + 3 \) and \( y < \frac{2}{3}x + 7 \)
In this case, the slopes are the same, but the second line is higher than the first. There could be points satisfying both inequalities since points can exist between the two lines. Thus, this option has solutions.
Therefore, the systems of inequalities with no solutions are found in Options #1 and #2, but since the question provides only one choice, and Option #1 is the only correct option that stands out clearly for producing no solutions (as it indicates a strict separation):
The system of inequalities in Option #1 has no solutions.