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To determine which points satisfy the inequality [tex]\( y < 0.5x + 2 \)[/tex], we need to evaluate this inequality for each point provided. Here are the steps:
1. Evaluate the inequality for [tex]\( (-3, -2) \)[/tex]:
- Compute [tex]\( 0.5 \times -3 + 2 \)[/tex].
[tex]\[ 0.5 \times (-3) + 2 = -1.5 + 2 = 0.5 \][/tex]
- Compare [tex]\( y = -2 \)[/tex] with [tex]\( 0.5 \)[/tex]:
[tex]\[ -2 < 0.5 \][/tex]
- Since this is true, the point [tex]\((-3, -2)\)[/tex] is a solution.
2. Evaluate the inequality for [tex]\( (-2, 1) \)[/tex]:
- Compute [tex]\( 0.5 \times -2 + 2 \)[/tex].
[tex]\[ 0.5 \times (-2) + 2 = -1 + 2 = 1 \][/tex]
- Compare [tex]\( y = 1 \)[/tex] with [tex]\( 1 \)[/tex]:
[tex]\[ 1 < 1 \][/tex]
- Since this is false, the point [tex]\((-2, 1)\)[/tex] is not a solution.
3. Evaluate the inequality for [tex]\( (-1, -2) \)[/tex]:
- Compute [tex]\( 0.5 \times -1 + 2 \)[/tex].
[tex]\[ 0.5 \times (-1) + 2 = -0.5 + 2 = 1.5 \][/tex]
- Compare [tex]\( y = -2 \)[/tex] with [tex]\( 1.5 \)[/tex]:
[tex]\[ -2 < 1.5 \][/tex]
- Since this is true, the point [tex]\((-1, -2)\)[/tex] is a solution.
4. Evaluate the inequality for [tex]\( (-1, 2) \)[/tex]:
- Compute [tex]\( 0.5 \times -1 + 2 \)[/tex].
[tex]\[ 0.5 \times (-1) + 2 = -0.5 + 2 = 1.5 \][/tex]
- Compare [tex]\( y = 2 \)[/tex] with [tex]\( 1.5 \)[/tex]:
[tex]\[ 2 < 1.5 \][/tex]
- Since this is false, the point [tex]\((-1, 2)\)[/tex] is not a solution.
5. Evaluate the inequality for [tex]\( (1, -2) \)[/tex]:
- Compute [tex]\( 0.5 \times 1 + 2 \)[/tex].
[tex]\[ 0.5 \times 1 + 2 = 0.5 + 2 = 2.5 \][/tex]
- Compare [tex]\( y = -2 \)[/tex] with [tex]\( 2.5 \)[/tex]:
[tex]\[ -2 < 2.5 \][/tex]
- Since this is true, the point [tex]\((1, -2)\)[/tex] is a solution.
Based on the evaluations, the points that satisfy the inequality [tex]\( y < 0.5x + 2 \)[/tex] are:
- [tex]\( (-3, -2) \)[/tex]
- [tex]\( (-1, -2) \)[/tex]
- [tex]\( (1, -2) \)[/tex]
Therefore, the three options to select are:
- [tex]\( (-3, -2) \)[/tex]
- [tex]\( (-1, -2) \)[/tex]
- [tex]\( (1, -2) \)[/tex]
1. Evaluate the inequality for [tex]\( (-3, -2) \)[/tex]:
- Compute [tex]\( 0.5 \times -3 + 2 \)[/tex].
[tex]\[ 0.5 \times (-3) + 2 = -1.5 + 2 = 0.5 \][/tex]
- Compare [tex]\( y = -2 \)[/tex] with [tex]\( 0.5 \)[/tex]:
[tex]\[ -2 < 0.5 \][/tex]
- Since this is true, the point [tex]\((-3, -2)\)[/tex] is a solution.
2. Evaluate the inequality for [tex]\( (-2, 1) \)[/tex]:
- Compute [tex]\( 0.5 \times -2 + 2 \)[/tex].
[tex]\[ 0.5 \times (-2) + 2 = -1 + 2 = 1 \][/tex]
- Compare [tex]\( y = 1 \)[/tex] with [tex]\( 1 \)[/tex]:
[tex]\[ 1 < 1 \][/tex]
- Since this is false, the point [tex]\((-2, 1)\)[/tex] is not a solution.
3. Evaluate the inequality for [tex]\( (-1, -2) \)[/tex]:
- Compute [tex]\( 0.5 \times -1 + 2 \)[/tex].
[tex]\[ 0.5 \times (-1) + 2 = -0.5 + 2 = 1.5 \][/tex]
- Compare [tex]\( y = -2 \)[/tex] with [tex]\( 1.5 \)[/tex]:
[tex]\[ -2 < 1.5 \][/tex]
- Since this is true, the point [tex]\((-1, -2)\)[/tex] is a solution.
4. Evaluate the inequality for [tex]\( (-1, 2) \)[/tex]:
- Compute [tex]\( 0.5 \times -1 + 2 \)[/tex].
[tex]\[ 0.5 \times (-1) + 2 = -0.5 + 2 = 1.5 \][/tex]
- Compare [tex]\( y = 2 \)[/tex] with [tex]\( 1.5 \)[/tex]:
[tex]\[ 2 < 1.5 \][/tex]
- Since this is false, the point [tex]\((-1, 2)\)[/tex] is not a solution.
5. Evaluate the inequality for [tex]\( (1, -2) \)[/tex]:
- Compute [tex]\( 0.5 \times 1 + 2 \)[/tex].
[tex]\[ 0.5 \times 1 + 2 = 0.5 + 2 = 2.5 \][/tex]
- Compare [tex]\( y = -2 \)[/tex] with [tex]\( 2.5 \)[/tex]:
[tex]\[ -2 < 2.5 \][/tex]
- Since this is true, the point [tex]\((1, -2)\)[/tex] is a solution.
Based on the evaluations, the points that satisfy the inequality [tex]\( y < 0.5x + 2 \)[/tex] are:
- [tex]\( (-3, -2) \)[/tex]
- [tex]\( (-1, -2) \)[/tex]
- [tex]\( (1, -2) \)[/tex]
Therefore, the three options to select are:
- [tex]\( (-3, -2) \)[/tex]
- [tex]\( (-1, -2) \)[/tex]
- [tex]\( (1, -2) \)[/tex]
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