Question

How many solutions does the system have?: y > - 1/3 x, y < - 1/3 x - 2

Answer 1

To determine the number of solutions for the given system of inequalities:

1. **First Inequality**: \( y > -\frac{1}{3}x \)
2. **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**.