To solve the problem, we need to analyze the equations of the sides of the triangle and determine the conditions under which the points \((\alpha, 0)\) and \((0, \beta)\) lie inside the triangle formed by the lines given.
### Step-by-Step Solution:
1. **Identify the Lines:**
The equations of the sides of the triangle are:
- Line AB: \(-x + y - 1 = 0\) (or \(y = x + 1\))
- Line BC: \(x + y - 1 = 0\) (or \(y = -x + 1\))
- Line AC: \(x = -4\)
2. **Find the Intersection Points:**
To find the vertices of the triangle, we need to find the intersection points of these lines.
- **Intersection of AB and AC:**
Substitute \(x = -4\) into \(y = x + 1\):
\[
y = -4 + 1 = -3 \quad \text{(Point A: } (-4, -3)\text{)}
\]
- **Intersection of BC and AC:**
Substitute \(x = -4\) into \(y = -x + 1\):
\[
y = -(-4) + 1 = 4 + 1 = 5 \quad \text{(Point B: } (-4, 5)\text{)}
\]
- **Intersection of AB and BC:**
Set \(x + 1 = -x + 1\):
\[
2x = 0 \implies x = 0
\]
Substitute \(x = 0\) into either line:
\[
y = 0 + 1 = 1 \quad \text{(Point C: } (0, 1)\text{)}
\]
3. **Plot the Triangle:**
The vertices of the triangle are:
- A: \((-4, -3)\)
- B: \((-4, 5)\)
- C: \((0, 1)\)
The triangle is formed in the coordinate plane.
4. **Determine Conditions for Points Inside the Triangle:**
The points \((\alpha, 0)\) and \((0, \beta)\) must lie inside the triangle.
- For \((\alpha, 0)\):
- It must be to the right of line AC (\(x = -4\)), so \(\alpha > -4\).
- It must be below line AB (\(y = x + 1\)), so:
\[
0 < \alpha + 1 \implies \alpha > -1
\]
- It must be below line BC (\(y = -x + 1\)), so:
\[
0 < -\alpha + 1 \implies \alpha < 1
\]
Combining these inequalities for \((\alpha, 0)\):
\[
-1 < \alpha < 1 \quad \text{and} \quad \alpha > -4 \quad \Rightarrow \quad -1 < \alpha < 1
\]
- For \((0, \beta)\):
- It must be above line AB (\(y = x + 1\)), so:
\[
\beta > 1
\]
- It must be below line BC (\(y = -x + 1\)), so:
\[
\beta < 1 \quad \text{(not possible)}
\]
Thus, \(\beta\) must be greater than 1.
5. **Conclusion:**
Since \(\alpha\) must be an integer, the possible values for \(\alpha\) are \(-1, 0\). For \(\beta\), since it must be greater than 1, the possible integer values are \(2, 3, 4, \ldots\).
