To determine if each point is a solution to the system of inequalities [tex]\( y \leq 3x + 4 \)[/tex] and [tex]\( y < 3x - 2 \)[/tex], we will check each point individually against both inequalities.
### For the point [tex]\((0, -2)\)[/tex]:
1. Check the first inequality [tex]\( y \leq 3x + 4 \)[/tex]
[tex]\[
-2 \leq 3(0) + 4
\][/tex]
[tex]\[
-2 \leq 4 \quad \text{(True)}
\][/tex]
2. Check the second inequality [tex]\( y < 3x - 2 \)[/tex]
[tex]\[
-2 < 3(0) - 2
\][/tex]
[tex]\[
-2 < -2 \quad \text{(False: because \(-2\) is not less than \(-2\))}
\][/tex]
Because the second inequality is not satisfied, [tex]\((0, -2)\)[/tex] is not a solution of the system.
[tex]\[
(0,-2) \; \square \; \text{(Not a solution)}
\][/tex]
### For the point [tex]\((5, 0)\)[/tex]:
1. Check the first inequality [tex]\( y \leq 3x + 4 \)[/tex]
[tex]\[
0 \leq 3(5) + 4
\][/tex]
[tex]\[
0 \leq 15 + 4
\][/tex]
[tex]\[
0 \leq 19 \quad \text{(True)}
\][/tex]
2. Check the second inequality [tex]\( y < 3x - 2 \)[/tex]
[tex]\[
0 < 3(5) - 2
\][/tex]
[tex]\[
0 < 15 - 2
\][/tex]
[tex]\[
0 < 13 \quad \text{(True)}
\][/tex]
Because both inequalities are satisfied, [tex]\((5,0)\)[/tex] is a solution of the system.
[tex]\[
(5,0) \; \checkmark \; \text{(Solution)}
\][/tex]
### For the point [tex]\((3, 13)\)[/tex]:
1. Check the first inequality [tex]\( y \leq 3x + 4 \)[/tex]
[tex]\[
13 \leq 3(3) + 4
\][/tex]
[tex]\[
13 \leq 9 + 4
\][/tex]
[tex]\[
13 \leq 13 \quad \text{(True)}
\][/tex]
2. Check the second inequality [tex]\( y < 3x - 2 \)[/tex]
[tex]\[
13 < 3(3) - 2
\][/tex]
[tex]\[
13 < 9 - 2
\][/tex]
[tex]\[
13 < 7 \quad \text{(False)}
\][/tex]
Because the second inequality is not satisfied, [tex]\((3, 13)\)[/tex] is not a solution of the system.
[tex]\[
(3,13) \; \square \; \text{(Not a solution)}
\][/tex]
In summary:
- [tex]\((0, -2)\)[/tex] is not a solution.
- [tex]\((5, 0)\)[/tex] is a solution.
- [tex]\((3, 13)\)[/tex] is not a solution.
[tex]\[
DONE
\][/tex]
