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For which system of inequalities is [tex]$(3,-7)$[/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 [tex]\((3, -7)\)[/tex] is a solution for, we will evaluate each system step-by-step.

### System A:
1. [tex]\(x + y < -4\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3 + (-7) = -4\)[/tex]
- Check: [tex]\(-4 < -4\)[/tex] (False)

2. [tex]\(3x + 2y < -5\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3(3) + 2(-7) = 9 - 14 = -5\)[/tex]
- Check: [tex]\(-5 < -5\)[/tex] (False)

Both inequalities must be true for the system to be true. Since the first condition is false, [tex]\( (3, -7) \)[/tex] does not satisfy system A.

### System B:
1. [tex]\(x + y \leq -4\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3 + (-7) = -4\)[/tex]
- Check: [tex]\(-4 \leq -4\)[/tex] (True)

2. [tex]\(3x + 2y < -5\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3(3) + 2(-7) = 9 - 14 = -5\)[/tex]
- Check: [tex]\(-5 < -5\)[/tex] (False)

Both inequalities must be true for the system to be true. Since the second condition is false, [tex]\( (3, -7) \)[/tex] does not satisfy system B.

### System C:
1. [tex]\(x + y < -4\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3 + (-7) = -4\)[/tex]
- Check: [tex]\(-4 < -4\)[/tex] (False)

2. [tex]\(3x + 2y \leq -5\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3(3) + 2(-7) = 9 - 14 = -5\)[/tex]
- Check: [tex]\(-5 \leq -5\)[/tex] (True)

Both inequalities must be true for the system to be true. Since the first condition is false, [tex]\( (3, -7) \)[/tex] does not satisfy system C.

### System D:
1. [tex]\(x + y \leq -4\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3 + (-7) = -4\)[/tex]
- Check: [tex]\(-4 \leq -4\)[/tex] (True)

2. [tex]\(3x + 2y \leq -5\)[/tex]
- Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex]
- [tex]\(3(3) + 2(-7) = 9 - 14 = -5\)[/tex]
- Check: [tex]\(-5 \leq -5\)[/tex] (True)

Both inequalities are true. Therefore, [tex]\((3, -7)\)[/tex] satisfies system D.

### Conclusion:
The point [tex]\((3, -7)\)[/tex] is a solution to system D.
So, the correct answer is:
D.
[tex]\[ \begin{array}{l} x+y \leq-4 \\ 3 x+2 y \leq-5 \end{array} \][/tex]
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