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].
