Which ordered pair makes both inequalities true?

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

A. \((0,0)\)
B. \((0,-1)\)
C. \((1,1)\)
D. [tex]\((3,0)\)[/tex]



Answer :

Let's solve the problem step-by-step to identify which ordered pair satisfies both inequalities:

The given inequalities are:
[tex]\[ y > -2x + 3 \][/tex]
[tex]\[ y \leq x - 2 \][/tex]

We need to check each ordered pair to see if it satisfies both inequalities.

1. For \( (0, 0) \):

- Check the first inequality:
[tex]\[ y > -2x + 3 \][/tex]
Substitute \( (0, 0) \) into the inequality:
[tex]\[ 0 > -2(0) + 3 \][/tex]
[tex]\[ 0 > 3 \][/tex]
This is false.

Since the first inequality is not satisfied, pair \( (0, 0) \) does not satisfy both inequalities.

2. For \( (0, -1) \):

- Check the first inequality:
[tex]\[ y > -2x + 3 \][/tex]
Substitute \( (0, -1) \) into the inequality:
[tex]\[ -1 > -2(0) + 3 \][/tex]
[tex]\[ -1 > 3 \][/tex]
This is false.

Since the first inequality is not satisfied, pair \( (0, -1) \) does not satisfy both inequalities.

3. For \( (1, 1) \):

- Check the first inequality:
[tex]\[ y > -2x + 3 \][/tex]
Substitute \( (1, 1) \) into the inequality:
[tex]\[ 1 > -2(1) + 3 \][/tex]
[tex]\[ 1 > 1 \][/tex]
This is false.

Since the first inequality is not satisfied, pair \( (1, 1) \) does not satisfy both inequalities.

4. For \( (3, 0) \):

- Check the first inequality:
[tex]\[ y > -2x + 3 \][/tex]
Substitute \( (3, 0) \) into the inequality:
[tex]\[ 0 > -2(3) + 3 \][/tex]
[tex]\[ 0 > -6 + 3 \][/tex]
[tex]\[ 0 > -3 \][/tex]
This is true.

- Check the second inequality:
[tex]\[ y \leq x - 2 \][/tex]
Substitute \( (3, 0) \) into the inequality:
[tex]\[ 0 \leq 3 - 2 \][/tex]
[tex]\[ 0 \leq 1 \][/tex]
This is true.

Since both inequalities are satisfied, pair \( (3, 0) \) does satisfy both inequalities.

Therefore, the ordered pair that makes both inequalities true is [tex]\( (3, 0) \)[/tex].