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For which system of inequalities is [tex][tex]$(3, -7)$[/tex][/tex] a solution?

A.
[tex]
\begin{array}{l}
x + y \ \textless \ -4 \\
3x + 2y \ \textless \ -5
\end{array}
[/tex]

B.
[tex]
\begin{array}{l}
x + y \leq -4 \\
3x + 2y \ \textless \ -5
\end{array}
[/tex]

C.
[tex]
\begin{array}{l}
x + y \ \textless \ -4 \\
3x + 2y \leq -5
\end{array}
[/tex]

D.
[tex]
\begin{array}{l}
x + y \leq -4 \\
3x + 2y \leq -5
\end{array}
[/tex]

Sagot :

GinnyAnswer
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]
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