Answer :
Let's determine which points satisfy the given linear inequality [tex]\( y < 0.5x + 2 \)[/tex].
1. For the point [tex]\((-3, -2)\)[/tex]:
- Substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = -2 \)[/tex] into the inequality:
[tex]\[ -2 < 0.5(-3) + 2 \][/tex]
[tex]\[ -2 < -1.5 + 2 \][/tex]
[tex]\[ -2 < 0.5 \quad \text{(True)} \][/tex]
- The point [tex]\((-3, -2)\)[/tex] satisfies the inequality.
2. For the point [tex]\((-2, 1)\)[/tex]:
- Substitute [tex]\( x = -2 \)[/tex] and [tex]\( y = 1 \)[/tex] into the inequality:
[tex]\[ 1 < 0.5(-2) + 2 \][/tex]
[tex]\[ 1 < -1 + 2 \][/tex]
[tex]\[ 1 < 1 \quad \text{(False)} \][/tex]
- The point [tex]\((-2, 1)\)[/tex] does not satisfy the inequality.
3. For the point [tex]\((-1, -2)\)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = -2 \)[/tex] into the inequality:
[tex]\[ -2 < 0.5(-1) + 2 \][/tex]
[tex]\[ -2 < -0.5 + 2 \][/tex]
[tex]\[ -2 < 1.5 \quad \text{(True)} \][/tex]
- The point [tex]\((-1, -2)\)[/tex] satisfies the inequality.
4. For the point [tex]\((-1, 2)\)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = 2 \)[/tex] into the inequality:
[tex]\[ 2 < 0.5(-1) + 2 \][/tex]
[tex]\[ 2 < -0.5 + 2 \][/tex]
[tex]\[ 2 < 1.5 \quad \text{(False)} \][/tex]
- The point [tex]\((-1, 2)\)[/tex] does not satisfy the inequality.
5. For the point [tex]\((1, -2)\)[/tex]:
- Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = -2 \)[/tex] into the inequality:
[tex]\[ -2 < 0.5(1) + 2 \][/tex]
[tex]\[ -2 < 0.5 + 2 \][/tex]
[tex]\[ -2 < 2.5 \quad \text{(True)} \][/tex]
- The point [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]
1. For the point [tex]\((-3, -2)\)[/tex]:
- Substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = -2 \)[/tex] into the inequality:
[tex]\[ -2 < 0.5(-3) + 2 \][/tex]
[tex]\[ -2 < -1.5 + 2 \][/tex]
[tex]\[ -2 < 0.5 \quad \text{(True)} \][/tex]
- The point [tex]\((-3, -2)\)[/tex] satisfies the inequality.
2. For the point [tex]\((-2, 1)\)[/tex]:
- Substitute [tex]\( x = -2 \)[/tex] and [tex]\( y = 1 \)[/tex] into the inequality:
[tex]\[ 1 < 0.5(-2) + 2 \][/tex]
[tex]\[ 1 < -1 + 2 \][/tex]
[tex]\[ 1 < 1 \quad \text{(False)} \][/tex]
- The point [tex]\((-2, 1)\)[/tex] does not satisfy the inequality.
3. For the point [tex]\((-1, -2)\)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = -2 \)[/tex] into the inequality:
[tex]\[ -2 < 0.5(-1) + 2 \][/tex]
[tex]\[ -2 < -0.5 + 2 \][/tex]
[tex]\[ -2 < 1.5 \quad \text{(True)} \][/tex]
- The point [tex]\((-1, -2)\)[/tex] satisfies the inequality.
4. For the point [tex]\((-1, 2)\)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = 2 \)[/tex] into the inequality:
[tex]\[ 2 < 0.5(-1) + 2 \][/tex]
[tex]\[ 2 < -0.5 + 2 \][/tex]
[tex]\[ 2 < 1.5 \quad \text{(False)} \][/tex]
- The point [tex]\((-1, 2)\)[/tex] does not satisfy the inequality.
5. For the point [tex]\((1, -2)\)[/tex]:
- Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = -2 \)[/tex] into the inequality:
[tex]\[ -2 < 0.5(1) + 2 \][/tex]
[tex]\[ -2 < 0.5 + 2 \][/tex]
[tex]\[ -2 < 2.5 \quad \text{(True)} \][/tex]
- The point [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]
