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Sagot :
To determine which system of inequalities the point [tex]\((3, -7)\)[/tex] satisfies, we need to substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into each inequality and check if the resulting statements are true.
Let's analyze each system one by one.
### Option A:
[tex]\[ \begin{cases} x + y < -4 \\ 3x + 2y < -5 \end{cases} \][/tex]
1. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(x + y < -4\)[/tex]:
[tex]\[ 3 + (-7) < -4 \implies -4 < -4 \quad \text{(False)} \][/tex]
2. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(3x + 2y < -5\)[/tex]:
[tex]\[ 3(3) + 2(-7) < -5 \implies 9 - 14 < -5 \implies -5 < -5 \quad \text{(False)} \][/tex]
Since both inequalities are not satisfied, Option A is not correct.
### Option B:
[tex]\[ \begin{cases} x + y \leq -4 \\ 3x + 2y < -5 \end{cases} \][/tex]
1. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(x + y \leq -4\)[/tex]:
[tex]\[ 3 + (-7) \leq -4 \implies -4 \leq -4 \quad \text{(True)} \][/tex]
2. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(3x + 2y < -5\)[/tex]:
[tex]\[ 3(3) + 2(-7) < -5 \implies 9 - 14 < -5 \implies -5 < -5 \quad \text{(False)} \][/tex]
Since one inequality is true but the other is false, Option B is not correct.
### Option C:
[tex]\[ \begin{cases} x + y < -4 \\ 3x + 2y \leq -5 \end{cases} \][/tex]
1. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(x + y < -4\)[/tex]:
[tex]\[ 3 + (-7) < -4 \implies -4 < -4 \quad \text{(False)} \][/tex]
2. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(3x + 2y \leq -5\)[/tex]:
[tex]\[ 3(3) + 2(-7) \leq -5 \implies 9 - 14 \leq -5 \implies -5 \leq -5 \quad \text{(True)} \][/tex]
Since one inequality is false, Option C is not correct.
### Option D:
[tex]\[ \begin{cases} x + y \leq -4 \\ 3x + 2y \leq -5 \end{cases} \][/tex]
1. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(x + y \leq -4\)[/tex]:
[tex]\[ 3 + (-7) \leq -4 \implies -4 \leq -4 \quad \text{(True)} \][/tex]
2. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(3x + 2y \leq -5\)[/tex]:
[tex]\[ 3(3) + 2(-7) \leq -5 \implies 9 - 14 \leq -5 \implies -5 \leq -5 \quad \text{(True)} \][/tex]
Since both inequalities are satisfied, Option D is correct.
Therefore, the system of inequalities that the point [tex]\((3, -7)\)[/tex] satisfies is:
[tex]\[ \boxed{D} \][/tex]
Let's analyze each system one by one.
### Option A:
[tex]\[ \begin{cases} x + y < -4 \\ 3x + 2y < -5 \end{cases} \][/tex]
1. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(x + y < -4\)[/tex]:
[tex]\[ 3 + (-7) < -4 \implies -4 < -4 \quad \text{(False)} \][/tex]
2. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(3x + 2y < -5\)[/tex]:
[tex]\[ 3(3) + 2(-7) < -5 \implies 9 - 14 < -5 \implies -5 < -5 \quad \text{(False)} \][/tex]
Since both inequalities are not satisfied, Option A is not correct.
### Option B:
[tex]\[ \begin{cases} x + y \leq -4 \\ 3x + 2y < -5 \end{cases} \][/tex]
1. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(x + y \leq -4\)[/tex]:
[tex]\[ 3 + (-7) \leq -4 \implies -4 \leq -4 \quad \text{(True)} \][/tex]
2. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(3x + 2y < -5\)[/tex]:
[tex]\[ 3(3) + 2(-7) < -5 \implies 9 - 14 < -5 \implies -5 < -5 \quad \text{(False)} \][/tex]
Since one inequality is true but the other is false, Option B is not correct.
### Option C:
[tex]\[ \begin{cases} x + y < -4 \\ 3x + 2y \leq -5 \end{cases} \][/tex]
1. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(x + y < -4\)[/tex]:
[tex]\[ 3 + (-7) < -4 \implies -4 < -4 \quad \text{(False)} \][/tex]
2. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(3x + 2y \leq -5\)[/tex]:
[tex]\[ 3(3) + 2(-7) \leq -5 \implies 9 - 14 \leq -5 \implies -5 \leq -5 \quad \text{(True)} \][/tex]
Since one inequality is false, Option C is not correct.
### Option D:
[tex]\[ \begin{cases} x + y \leq -4 \\ 3x + 2y \leq -5 \end{cases} \][/tex]
1. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(x + y \leq -4\)[/tex]:
[tex]\[ 3 + (-7) \leq -4 \implies -4 \leq -4 \quad \text{(True)} \][/tex]
2. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into [tex]\(3x + 2y \leq -5\)[/tex]:
[tex]\[ 3(3) + 2(-7) \leq -5 \implies 9 - 14 \leq -5 \implies -5 \leq -5 \quad \text{(True)} \][/tex]
Since both inequalities are satisfied, Option D is correct.
Therefore, the system of inequalities that the point [tex]\((3, -7)\)[/tex] satisfies is:
[tex]\[ \boxed{D} \][/tex]
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