To analyze the given inequalities:
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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.
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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 \((-3, -5)\):
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First Inequality: \( -(-3) + (-5) < 3 \) \[ 3 - 5 < 3 \implies -2 < 3 \quad (\text{True}) \]
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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 \((1, -1)\):
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First Inequality: \( -1 < 1 + 3 \) \[ -1 < 4 \quad (\text{True}) \]
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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.