Question

The graph is: -x + y < 3/1 , y ≥ -x + - 3/1. What is the inequality symbol of the dotted line? What is the inequality symbol of the solid line? Is the point (-3, -5) a solution to the system of inequalities? Is the point (1, -1) a solution to the system of inequalities?

Answer 1

To analyze the given inequalities:

1. **Inequality 1:** \( -x + y < 3 \)
- This can be rewritten as \( y < x + 3 \).
- The corresponding line \( y = x + 3 \) represents the boundary, and since the inequality is strict (less than), the graph of this line will be a **dotted line**.

2. **Inequality 2:** \( y \geq -x - 3 \)
- The associated line \( y = -x - 3 \) serves as the boundary for this inequality.
- Since this inequality includes equal to (greater than or equal to), the graph of this line will be a **solid line**.

Next, to determine whether the points are solutions to the system of inequalities, we need to test each point against both inequalities.

### Checking Point \((-3, -5)\):
1. **First Inequality:** \( -(-3) + (-5) < 3 \)
\[
3 - 5 < 3 \implies -2 < 3 \quad (\text{True})
\]

2. **Second Inequality:** \(-5 \geq -(-3) - 3\)
\[
-5 \geq 3 - 3 \implies -5 \geq 0 \quad (\text{False})
\]

Since the second inequality is false for the point \((-3, -5)\), this point is **not a solution** to the system.

### Checking Point \((1, -1)\):
1. **First Inequality:** \( -1 < 1 + 3 \)
\[
-1 < 4 \quad (\text{True})
\]

2. **Second Inequality:** \( -1 \geq -1 - 3 \)
\[
-1 \geq -4 \quad (\text{True})
\]

Since both inequalities are true for the point \((1, -1)\), this point **is a solution** to the system.

### Summary:
- The inequality symbol for the dotted line: **< (less than)**
- The inequality symbol for the solid line: **≥ (greater than or equal to)**
- Point \((-3, -5)\) is **not a solution**.
- Point \((1, -1)\) is **a solution**.