Answer :
To determine which of the given points satisfy the system of inequalities, we need to check each point against the inequalities:
[tex]\[ \begin{array}{l} y \leq -2x + 6 \\ x > 1 \end{array} \][/tex]
Let's go through each point one by one:
### Point A: [tex]\((1, 4)\)[/tex]
1. Check [tex]\(x > 1\)[/tex]:
- [tex]\(x = 1\)[/tex]; since [tex]\(1 \leq 1\)[/tex], this point does not satisfy the inequality [tex]\(x > 1\)[/tex].
Since it fails the second inequality, point A does not satisfy the system of inequalities.
### Point B: [tex]\((2, 1)\)[/tex]
1. Check [tex]\(x > 1\)[/tex]:
- [tex]\(x = 2\)[/tex]; since [tex]\(2 > 1\)[/tex], this point satisfies the inequality [tex]\(x > 1\)[/tex].
2. Check [tex]\(y \leq -2x + 6\)[/tex]:
- Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 1\)[/tex]: [tex]\(1 \leq -2(2) + 6\)[/tex] which simplifies to [tex]\(1 \leq 2\)[/tex].
Since it satisfies both inequalities, point B does satisfy the system of inequalities.
### Point C: [tex]\((0, 6)\)[/tex]
1. Check [tex]\(x > 1\)[/tex]:
- [tex]\(x = 0\)[/tex]; since [tex]\(0 \leq 1\)[/tex], this point does not satisfy the inequality [tex]\(x > 1\)[/tex].
Since it fails the second inequality, point C does not satisfy the system of inequalities.
### Point D: [tex]\((2, 11)\)[/tex]
1. Check [tex]\(x > 1\)[/tex]:
- [tex]\(x = 2\)[/tex]; since [tex]\(2 > 1\)[/tex], this point satisfies the inequality [tex]\(x > 1\)[/tex].
2. Check [tex]\(y \leq -2x + 6\)[/tex]:
- Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 11\)[/tex]: [tex]\(11 \leq -2(2) + 6\)[/tex] which simplifies to [tex]\(11 \leq 2\)[/tex].
Since it fails the first inequality, point D does not satisfy the system of inequalities.
### Point E: [tex]\((5, -6)\)[/tex]
1. Check [tex]\(x > 1\)[/tex]:
- [tex]\(x = 5\)[/tex]; since [tex]\(5 > 1\)[/tex], this point satisfies the inequality [tex]\(x > 1\)[/tex].
2. Check [tex]\(y \leq -2x + 6\)[/tex]:
- Substitute [tex]\(x = 5\)[/tex] and [tex]\(y = -6\)[/tex]: [tex]\(-6 \leq -2(5) + 6\)[/tex] which simplifies to [tex]\(-6 \leq -4\)[/tex].
Since it satisfies both inequalities, point E does satisfy the system of inequalities.
### Conclusion
The points that satisfy the system of inequalities are:
- B. [tex]\((2, 1)\)[/tex]
- E. [tex]\((5, -6)\)[/tex]
Thus, the solutions to the system of inequalities are:
[tex]\[ \{(2, 1), (5, -6)\} \][/tex]
[tex]\[ \begin{array}{l} y \leq -2x + 6 \\ x > 1 \end{array} \][/tex]
Let's go through each point one by one:
### Point A: [tex]\((1, 4)\)[/tex]
1. Check [tex]\(x > 1\)[/tex]:
- [tex]\(x = 1\)[/tex]; since [tex]\(1 \leq 1\)[/tex], this point does not satisfy the inequality [tex]\(x > 1\)[/tex].
Since it fails the second inequality, point A does not satisfy the system of inequalities.
### Point B: [tex]\((2, 1)\)[/tex]
1. Check [tex]\(x > 1\)[/tex]:
- [tex]\(x = 2\)[/tex]; since [tex]\(2 > 1\)[/tex], this point satisfies the inequality [tex]\(x > 1\)[/tex].
2. Check [tex]\(y \leq -2x + 6\)[/tex]:
- Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 1\)[/tex]: [tex]\(1 \leq -2(2) + 6\)[/tex] which simplifies to [tex]\(1 \leq 2\)[/tex].
Since it satisfies both inequalities, point B does satisfy the system of inequalities.
### Point C: [tex]\((0, 6)\)[/tex]
1. Check [tex]\(x > 1\)[/tex]:
- [tex]\(x = 0\)[/tex]; since [tex]\(0 \leq 1\)[/tex], this point does not satisfy the inequality [tex]\(x > 1\)[/tex].
Since it fails the second inequality, point C does not satisfy the system of inequalities.
### Point D: [tex]\((2, 11)\)[/tex]
1. Check [tex]\(x > 1\)[/tex]:
- [tex]\(x = 2\)[/tex]; since [tex]\(2 > 1\)[/tex], this point satisfies the inequality [tex]\(x > 1\)[/tex].
2. Check [tex]\(y \leq -2x + 6\)[/tex]:
- Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 11\)[/tex]: [tex]\(11 \leq -2(2) + 6\)[/tex] which simplifies to [tex]\(11 \leq 2\)[/tex].
Since it fails the first inequality, point D does not satisfy the system of inequalities.
### Point E: [tex]\((5, -6)\)[/tex]
1. Check [tex]\(x > 1\)[/tex]:
- [tex]\(x = 5\)[/tex]; since [tex]\(5 > 1\)[/tex], this point satisfies the inequality [tex]\(x > 1\)[/tex].
2. Check [tex]\(y \leq -2x + 6\)[/tex]:
- Substitute [tex]\(x = 5\)[/tex] and [tex]\(y = -6\)[/tex]: [tex]\(-6 \leq -2(5) + 6\)[/tex] which simplifies to [tex]\(-6 \leq -4\)[/tex].
Since it satisfies both inequalities, point E does satisfy the system of inequalities.
### Conclusion
The points that satisfy the system of inequalities are:
- B. [tex]\((2, 1)\)[/tex]
- E. [tex]\((5, -6)\)[/tex]
Thus, the solutions to the system of inequalities are:
[tex]\[ \{(2, 1), (5, -6)\} \][/tex]
