To determine which ordered pairs are solutions to the given system of inequalities, we need to check each pair against all the inequalities in the system:
The system of inequalities is:
1. [tex]\( y \geq 3x - 5 \)[/tex]
2. [tex]\( y \leq x + 10 \)[/tex]
3. [tex]\( y \geq -2x + 1 \)[/tex]
We will test each ordered pair to see if it satisfies all three inequalities.
### Testing Pair [tex]\((2, 0)\)[/tex]:
1. First inequality: [tex]\(0 \geq 3(2) - 5\)[/tex]
[tex]\[
0 \geq 6 - 5 \implies 0 \geq 1 \quad \text{(False)}
\][/tex]
Since the first inequality is not satisfied, [tex]\((2, 0)\)[/tex] is not a solution.
### Testing Pair [tex]\((-12, 20)\)[/tex]:
1. First inequality: [tex]\(20 \geq 3(-12) - 5\)[/tex]
[tex]\[
20 \geq -36 - 5 \implies 20 \geq -41 \quad \text{(True)}
\][/tex]
2. Second inequality: [tex]\(20 \leq -12 + 10\)[/tex]
[tex]\[
20 \leq -2 \quad \text{(False)}
\][/tex]
Since the second inequality is not satisfied, [tex]\((-12, 20)\)[/tex] is not a solution.
### Testing Pair [tex]\((10, -10)\)[/tex]:
1. First inequality: [tex]\(-10 \geq 3(10) - 5\)[/tex]
[tex]\[
-10 \geq 30 - 5 \implies -10 \geq 25 \quad \text{(False)}
\][/tex]
Since the first inequality is not satisfied, [tex]\((10, -10)\)[/tex] is not a solution.
### Testing Pair [tex]\((10, 10)\)[/tex]:
1. First inequality: [tex]\(10 \geq 3(10) - 5\)[/tex]
[tex]\[
10 \geq 30 - 5 \implies 10 \geq 25 \quad \text{(False)}
\][/tex]
Since the first inequality is not satisfied, [tex]\((10, 10)\)[/tex] is not a solution.
### Conclusion:
After checking all the ordered pairs against the system of inequalities, none of the given pairs satisfy all three inequalities. Therefore, none of the ordered pairs [tex]\((2, 0)\)[/tex], [tex]\((-12, 20)\)[/tex], [tex]\((10, -10)\)[/tex], and [tex]\((10, 10)\)[/tex] are solutions to the given system of inequalities.
Thus, the final results are:
[tex]\[
(False, False, False, False)
\][/tex]
