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The graph below represents the following system of inequalities:

[tex]\[
\begin{array}{l}
y \ \textgreater \ -3x + 5 \\
y \ \textgreater \ x - 2
\end{array}
\][/tex]

Which point [tex]\((x, y)\)[/tex] satisfies the given system of inequalities?

A. [tex]\((3, -2)\)[/tex]

B. [tex]\((4, 1)\)[/tex]

C. [tex]\((-1, 4)\)[/tex]

D. [tex]\((2, 3)\)[/tex]

Sagot :

GinnyAnsweridn
Let's analyze the system of inequalities provided:

1. [tex]\( y > -3x + 5 \)[/tex]
2. [tex]\( y > x - 2 \)[/tex]

We are given four points, and we need to determine which of these points satisfies both inequalities. Let's check each point one by one:

### Point A: [tex]\((3, -2)\)[/tex]

1. For the first inequality [tex]\( y > -3x + 5 \)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = -2 \)[/tex]:
[tex]\[ -2 > -3(3) + 5 \][/tex]
[tex]\[ -2 > -9 + 5 \][/tex]
[tex]\[ -2 > -4 \][/tex]

This is a true statement.

2. For the second inequality [tex]\( y > x - 2 \)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = -2 \)[/tex]:
[tex]\[ -2 > 3 - 2 \][/tex]
[tex]\[ -2 > 1 \][/tex]

This is a false statement.

Since Point A does not satisfy both inequalities, it is not a solution.

### Point B: [tex]\((4, 1)\)[/tex]

1. For the first inequality [tex]\( y > -3x + 5 \)[/tex]:
- Substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[ 1 > -3(4) + 5 \][/tex]
[tex]\[ 1 > -12 + 5 \][/tex]
[tex]\[ 1 > -7 \][/tex]

This is a true statement.

2. For the second inequality [tex]\( y > x - 2 \)[/tex]:
- Substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[ 1 > 4 - 2 \][/tex]
[tex]\[ 1 > 2 \][/tex]

This is a false statement.

Since Point B does not satisfy both inequalities, it is not a solution.

### Point C: [tex]\((-1, 4)\)[/tex]

1. For the first inequality [tex]\( y > -3x + 5 \)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = 4 \)[/tex]:
[tex]\[ 4 > -3(-1) + 5 \][/tex]
[tex]\[ 4 > 3 + 5 \][/tex]
[tex]\[ 4 > 8 \][/tex]

This is a false statement.

2. For the second inequality [tex]\( y > x - 2 \)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = 4 \)[/tex]:
[tex]\[ 4 > -1 - 2 \][/tex]
[tex]\[ 4 > -3 \][/tex]

This is a true statement.

Since Point C does not satisfy both inequalities, it is not a solution.

### Point D: [tex]\((2, 3)\)[/tex]

1. For the first inequality [tex]\( y > -3x + 5 \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[ 3 > -3(2) + 5 \][/tex]
[tex]\[ 3 > -6 + 5 \][/tex]
[tex]\[ 3 > -1 \][/tex]

This is a true statement.

2. For the second inequality [tex]\( y > x - 2 \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[ 3 > 2 - 2 \][/tex]
[tex]\[ 3 > 0 \][/tex]

This is also a true statement.

Since Point D satisfies both inequalities, it is the solution.

Therefore, the correct answer is:
D. [tex]\((2, 3)\)[/tex]
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