To determine which points satisfy the linear inequality [tex]\( y < 0.5x + 2 \)[/tex], we'll analyze each of the given points one by one. Let's go through each point and check if it meets the inequality condition.
1. Point [tex]\((-3, -2)\)[/tex]:
- Substitute [tex]\( x = -3 \)[/tex] into the inequality to calculate the right-hand side:
[tex]\[ 0.5 \cdot (-3) + 2 = -1.5 + 2 = 0.5 \][/tex]
- Check if the y-coordinate of the point [tex]\((-2)\)[/tex] is less than [tex]\(0.5\)[/tex]:
[tex]\[ -2 < 0.5 \][/tex]
- This is true, so [tex]\((-3, -2)\)[/tex] satisfies the inequality.
2. Point [tex]\((-2, 1)\)[/tex]:
- Substitute [tex]\( x = -2 \)[/tex] into the inequality to calculate the right-hand side:
[tex]\[ 0.5 \cdot (-2) + 2 = -1 + 2 = 1 \][/tex]
- Check if the y-coordinate of the point [tex]\((1)\)[/tex] is less than [tex]\(1\)[/tex]:
[tex]\[ 1 < 1 \][/tex]
- This is false, so [tex]\((-2, 1)\)[/tex] does not satisfy the inequality.
3. Point [tex]\((-1, -2)\)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] into the inequality to calculate the right-hand side:
[tex]\[ 0.5 \cdot (-1) + 2 = -0.5 + 2 = 1.5 \][/tex]
- Check if the y-coordinate of the point [tex]\((-2)\)[/tex] is less than [tex]\(1.5\)[/tex]:
[tex]\[ -2 < 1.5 \][/tex]
- This is true, so [tex]\((-1, -2)\)[/tex] satisfies the inequality.
4. Point [tex]\((-1, 2)\)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] into the inequality to calculate the right-hand side:
[tex]\[ 0.5 \cdot (-1) + 2 = -0.5 + 2 = 1.5 \][/tex]
- Check if the y-coordinate of the point [tex]\((2)\)[/tex] is less than [tex]\(1.5\)[/tex]:
[tex]\[ 2 < 1.5 \][/tex]
- This is false, so [tex]\((-1, 2)\)[/tex] does not satisfy the inequality.
5. Point [tex]\((1, -2)\)[/tex]:
- Substitute [tex]\( x = 1 \)[/tex] into the inequality to calculate the right-hand side:
[tex]\[ 0.5 \cdot (1) + 2 = 0.5 + 2 = 2.5 \][/tex]
- Check if the y-coordinate of the point [tex]\((-2)\)[/tex] is less than [tex]\(2.5\)[/tex]:
[tex]\[ -2 < 2.5 \][/tex]
- This is true, so [tex]\((1, -2)\)[/tex] satisfies the inequality.
Therefore, the three points that satisfy the inequality [tex]\( y < 0.5x + 2 \)[/tex] are:
- [tex]\((-3, -2)\)[/tex]
- [tex]\((-1, -2)\)[/tex]
- [tex]\((1, -2)\)[/tex]
So, the correct options are:
1. [tex]\((-3, -2)\)[/tex]
2. [tex]\((-1, -2)\)[/tex]
3. [tex]\((1, -2)\)[/tex]
