To determine which points satisfy the system of inequalities:
[tex]\[
\begin{array}{l}
2x + 3y \geq 6 \\
3x + 2y \leq 6
\end{array}
\][/tex]
we need to check each point given against both inequalities.
### Checking Point (1.5, 1)
1. First Inequality: [tex]\(2x + 3y \geq 6 \)[/tex]
[tex]\[
2(1.5) + 3(1) = 3 + 3 = 6 \quad \text{(satisfies \( \geq 6 \))}
\][/tex]
2. Second Inequality: [tex]\(3x + 2y \leq 6 \)[/tex]
[tex]\[
3(1.5) + 2(1) = 4.5 + 2 = 6.5 \quad \text{(does not satisfy \( \leq 6 \))}
\][/tex]
Point (1.5, 1) does not satisfy the system.
### Checking Point (0.5, 2)
1. First Inequality: [tex]\(2x + 3y \geq 6 \)[/tex]
[tex]\[
2(0.5) + 3(2) = 1 + 6 = 7 \quad \text{(satisfies \( \geq 6 \))}
\][/tex]
2. Second Inequality: [tex]\(3x + 2y \leq 6 \)[/tex]
[tex]\[
3(0.5) + 2(2) = 1.5 + 4 = 5.5 \quad \text{(satisfies \( \leq 6 \))}
\][/tex]
Point (0.5, 2) satisfies the system.
### Checking Point (-1, 2.5)
1. First Inequality: [tex]\(2x + 3y \geq 6 \)[/tex]
[tex]\[
2(-1) + 3(2.5) = -2 + 7.5 = 5.5 \quad \text{(does not satisfy \( \geq 6 \))}
\][/tex]
Point (-1, 2.5) does not satisfy the system (no need to check the second inequality after the first fails).
### Checking Point (-2, 4)
1. First Inequality: [tex]\(2x + 3y \geq 6 \)[/tex]
[tex]\[
2(-2) + 3(4) = -4 + 12 = 8 \quad \text{(satisfies \( \geq 6 \))}
\][/tex]
2. Second Inequality: [tex]\(3x + 2y \leq 6 \)[/tex]
[tex]\[
3(-2) + 2(4) = -6 + 8 = 2 \quad \text{(satisfies \( \leq 6 \))}
\][/tex]
Point (-2, 4) satisfies the system.
### Conclusion
The points that satisfy the system of inequalities are:
[tex]\[
(0.5, 2) \text{ and } (-2, 4)
\][/tex]
