Answer :
To determine which points are solutions to the inequality [tex]\( y < 0.5x + 2 \)[/tex], let's check each point one by one:
1. Point [tex]\((-3, -2)\)[/tex]:
[tex]\[ y = -2 \quad \text{and} \quad 0.5 \cdot (-3) + 2 = -1.5 + 2 = 0.5 \][/tex]
We compare [tex]\( y \)[/tex] and [tex]\( 0.5x + 2 \)[/tex]:
[tex]\[ -2 < 0.5 \][/tex]
This inequality is true, so [tex]\((-3, -2)\)[/tex] is a solution.
2. Point [tex]\((-2, 1)\)[/tex]:
[tex]\[ y = 1 \quad \text{and} \quad 0.5 \cdot (-2) + 2 = -1 + 2 = 1 \][/tex]
We compare [tex]\( y \)[/tex] and [tex]\( 0.5x + 2 \)[/tex]:
[tex]\[ 1 < 1 \][/tex]
This inequality is false, so [tex]\((-2, 1)\)[/tex] is not a solution.
3. Point [tex]\((-1, -2)\)[/tex]:
[tex]\[ y = -2 \quad \text{and} \quad 0.5 \cdot (-1) + 2 = -0.5 + 2 = 1.5 \][/tex]
We compare [tex]\( y \)[/tex] and [tex]\( 0.5x + 2 \)[/tex]:
[tex]\[ -2 < 1.5 \][/tex]
This inequality is true, so [tex]\((-1, -2)\)[/tex] is a solution.
4. Point [tex]\((-1, 2)\)[/tex]:
[tex]\[ y = 2 \quad \text{and} \quad 0.5 \cdot (-1) + 2 = -0.5 + 2 = 1.5 \][/tex]
We compare [tex]\( y \)[/tex] and [tex]\( 0.5x + 2 \)[/tex]:
[tex]\[ 2 < 1.5 \][/tex]
This inequality is false, so [tex]\((-1, 2)\)[/tex] is not a solution.
5. Point [tex]\((1, -2)\)[/tex]:
[tex]\[ y = -2 \quad \text{and} \quad 0.5 \cdot 1 + 2 = 0.5 + 2 = 2.5 \][/tex]
We compare [tex]\( y \)[/tex] and [tex]\( 0.5x + 2 \)[/tex]:
[tex]\[ -2 < 2.5 \][/tex]
This inequality is true, so [tex]\((1, -2)\)[/tex] is a solution.
Therefore, the points that are solutions to the inequality [tex]\( y < 0.5x + 2 \)[/tex] are:
- [tex]\((-3, -2)\)[/tex]
- [tex]\((-1, -2)\)[/tex]
- [tex]\((1, -2)\)[/tex]
Thus, the correct three options are:
[tex]\[ \boxed{(-3,-2)} \boxed{(-1,-2)} \boxed{(1,-2)} \][/tex]
1. Point [tex]\((-3, -2)\)[/tex]:
[tex]\[ y = -2 \quad \text{and} \quad 0.5 \cdot (-3) + 2 = -1.5 + 2 = 0.5 \][/tex]
We compare [tex]\( y \)[/tex] and [tex]\( 0.5x + 2 \)[/tex]:
[tex]\[ -2 < 0.5 \][/tex]
This inequality is true, so [tex]\((-3, -2)\)[/tex] is a solution.
2. Point [tex]\((-2, 1)\)[/tex]:
[tex]\[ y = 1 \quad \text{and} \quad 0.5 \cdot (-2) + 2 = -1 + 2 = 1 \][/tex]
We compare [tex]\( y \)[/tex] and [tex]\( 0.5x + 2 \)[/tex]:
[tex]\[ 1 < 1 \][/tex]
This inequality is false, so [tex]\((-2, 1)\)[/tex] is not a solution.
3. Point [tex]\((-1, -2)\)[/tex]:
[tex]\[ y = -2 \quad \text{and} \quad 0.5 \cdot (-1) + 2 = -0.5 + 2 = 1.5 \][/tex]
We compare [tex]\( y \)[/tex] and [tex]\( 0.5x + 2 \)[/tex]:
[tex]\[ -2 < 1.5 \][/tex]
This inequality is true, so [tex]\((-1, -2)\)[/tex] is a solution.
4. Point [tex]\((-1, 2)\)[/tex]:
[tex]\[ y = 2 \quad \text{and} \quad 0.5 \cdot (-1) + 2 = -0.5 + 2 = 1.5 \][/tex]
We compare [tex]\( y \)[/tex] and [tex]\( 0.5x + 2 \)[/tex]:
[tex]\[ 2 < 1.5 \][/tex]
This inequality is false, so [tex]\((-1, 2)\)[/tex] is not a solution.
5. Point [tex]\((1, -2)\)[/tex]:
[tex]\[ y = -2 \quad \text{and} \quad 0.5 \cdot 1 + 2 = 0.5 + 2 = 2.5 \][/tex]
We compare [tex]\( y \)[/tex] and [tex]\( 0.5x + 2 \)[/tex]:
[tex]\[ -2 < 2.5 \][/tex]
This inequality is true, so [tex]\((1, -2)\)[/tex] is a solution.
Therefore, the points that are solutions to the inequality [tex]\( y < 0.5x + 2 \)[/tex] are:
- [tex]\((-3, -2)\)[/tex]
- [tex]\((-1, -2)\)[/tex]
- [tex]\((1, -2)\)[/tex]
Thus, the correct three options are:
[tex]\[ \boxed{(-3,-2)} \boxed{(-1,-2)} \boxed{(1,-2)} \][/tex]
