To determine the number of solutions for the given system of inequalities:
- First Inequality: \( y > -\frac{1}{3}x \)
- Second Inequality: \( y < -\frac{1}{3}x - 2 \)
We will analyze these inequalities graphically.
Step 1: Graph the first inequality \( y = -\frac{1}{3}x \)
This line has a slope of \(-\frac{1}{3}\) and a y-intercept of \(0\). The area above this line represents the solutions for the first inequality.
Step 2: Graph the second inequality \( y = -\frac{1}{3}x - 2 \)
This line also has a slope of \(-\frac{1}{3}\) but a y-intercept of \(-2\). The area below this line represents the solutions for the second inequality.
Step 3: Analyze the intersection of the two regions
Now we check whether there are any overlapping regions:
- The first inequality shows that \( y \) must be above the line \( y = -\frac{1}{3}x \).
- The second inequality shows that \( y \) must be below the line \( y = -\frac{1}{3}x - 2 \).
Since both lines are parallel (they have the same slope), and the line defined by the second inequality is always 2 units lower than the line defined by the first inequality, there can be no \( y \) values that satisfy both inequalities simultaneously.
Conclusion: There are no overlapping regions for the two inequalities, thus the system has no solutions.