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.