Let's examine each point to determine if it satisfies the inequality [tex]\( y < 0.5x + 2 \)[/tex].
1. 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]\[
-2 < 0.5
\][/tex]
This inequality is true. Therefore, [tex]\((-3, -2)\)[/tex] is a solution.
2. 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]\[
1 < 1
\][/tex]
This inequality is not true because 1 is not less than 1. Therefore, [tex]\((-2, 1)\)[/tex] is not a solution.
3. 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]\[
-2 < 1.5
\][/tex]
This inequality is true. Therefore, [tex]\((-1, -2)\)[/tex] is a solution.
4. 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]\[
2 < 1.5
\][/tex]
This inequality is not true because 2 is not less than 1.5. Therefore, [tex]\((-1, 2)\)[/tex] is not a solution.
5. 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]\[
-2 < 2.5
\][/tex]
This inequality is true. Therefore, [tex]\((1, -2)\)[/tex] is a solution.
Based on our analysis, 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]
