To determine which points are solutions to the linear inequality [tex]\( y < 0.5x + 2 \)[/tex], we need to test each given point to see if it satisfies this inequality.
Let's check each point step-by-step:
1. Point [tex]\((-3, -2)\)[/tex]:
[tex]\[
y = -2
\][/tex]
[tex]\[
0.5x + 2 = 0.5(-3) + 2 = -1.5 + 2 = 0.5
\][/tex]
We need to check if [tex]\( y < 0.5 \)[/tex]:
[tex]\[
-2 < 0.5 \quad \text{(True)}
\][/tex]
So, [tex]\((-3, -2)\)[/tex] is a solution.
2. Point [tex]\((-2, 1)\)[/tex]:
[tex]\[
y = 1
\][/tex]
[tex]\[
0.5x + 2 = 0.5(-2) + 2 = -1 + 2 = 1
\][/tex]
We need to check if [tex]\( y < 1 \)[/tex]:
[tex]\[
1 < 1 \quad \text{(False)}
\][/tex]
So, [tex]\((-2, 1)\)[/tex] is not a solution.
3. Point [tex]\((-1, -2)\)[/tex]:
[tex]\[
y = -2
\][/tex]
[tex]\[
0.5x + 2 = 0.5(-1) + 2 = -0.5 + 2 = 1.5
\][/tex]
We need to check if [tex]\( y < 1.5 \)[/tex]:
[tex]\[
-2 < 1.5 \quad \text{(True)}
\][/tex]
So, [tex]\((-1, -2)\)[/tex] is a solution.
4. Point [tex]\((-1, 2)\)[/tex]:
[tex]\[
y = 2
\][/tex]
[tex]\[
0.5x + 2 = 0.5(-1) + 2 = -0.5 + 2 = 1.5
\][/tex]
We need to check if [tex]\( y < 1.5 \)[/tex]:
[tex]\[
2 < 1.5 \quad \text{(False)}
\][/tex]
So, [tex]\((-1, 2)\)[/tex] is not a solution.
5. Point [tex]\((1, -2)\)[/tex]:
[tex]\[
y = -2
\][/tex]
[tex]\[
0.5x + 2 = 0.5(1) + 2 = 0.5 + 2 = 2.5
\][/tex]
We need to check if [tex]\( y < 2.5 \)[/tex]:
[tex]\[
-2 < 2.5 \quad \text{(True)}
\][/tex]
So, [tex]\((1, -2)\)[/tex] is a solution.
Therefore, the points that satisfy the inequality [tex]\( y < 0.5x + 2 \)[/tex] are: [tex]\((-3, -2)\)[/tex], [tex]\((-1, -2)\)[/tex], and [tex]\((1, -2)\)[/tex].
