To analyze the system of inequalities given:
-
Blue inequality: \( y > 3x + 1 \)
- This represents a region above the line \( y = 3x + 1 \).
- The line itself is not included in the solution since the inequality is strict (greater than).
-
Red inequality: \( y < 3x - 6 \)
- This represents a region below the line \( y = 3x - 6 \).
- Similarly, the line itself is not included since the inequality is also strict (less than).
Next, let's understand the behavior of these two lines:
- The line \( y = 3x + 1 \) has a slope of 3 and a y-intercept of 1.
- The line \( y = 3x - 6 \) also has a slope of 3 but a y-intercept of -6.
Analyzing the Graph of the Lines:
Both lines have the same slope (3), which means they are parallel. Since both have different y-intercepts (1 and -6), they will never intersect.
Number of Solutions:
Since there is no point at which both inequalities can be true (since there is no intersection), we cannot find any points that satisfy both inequalities simultaneously. Therefore, the solution set is empty.
Conclusion:
In graphical terms, the purple region, which represents the overlapping area between the blue and red shaded regions, is empty. Thus, the answer choice that represents the graph of the system and indicates the number of solutions is:
No solutions (empty region).