To determine if each ordered pair is a solution to the system of inequalities \( y < 2x + 2 \) and \( y \geq -3x + 4 \), we will substitute each ordered pair into both inequalities and check if they satisfy both conditions.
Inequalities:
- \( y < 2x + 2 \)
- \( y \geq -3x + 4 \)
Check each ordered pair:
-
For (1, 4):
- Substitute \( x = 1 \) and \( y = 4 \):
- Check 1: \( 4 < 2(1) + 2 \) → \( 4 < 4 \) (False)
- Check 2: \( 4 \geq -3(1) + 4 \) → \( 4 \geq 1 \) (True)
- Result: Not a solution (fails first inequality)
- Substitute \( x = 1 \) and \( y = 4 \):
-
For (2, 2):
- Substitute \( x = 2 \) and \( y = 2 \):
- Check 1: \( 2 < 2(2) + 2 \) → \( 2 < 6 \) (True)
- Check 2: \( 2 \geq -3(2) + 4 \) → \( 2 \geq -2 \) (True)
- Result: It is a solution (satisfies both inequalities)
- Substitute \( x = 2 \) and \( y = 2 \):
-
For (1, 1):
- Substitute \( x = 1 \) and \( y = 1 \):
- Check 1: \( 1 < 2(1) + 2 \) → \( 1 < 4 \) (True)
- Check 2: \( 1 \geq -3(1) + 4 \) → \( 1 \geq 1 \) (True)
- Result: It is a solution (satisfies both inequalities)
- Substitute \( x = 1 \) and \( y = 1 \):
-
For (0, 4):
- Substitute \( x = 0 \) and \( y = 4 \):
- Check 1: \( 4 < 2(0) + 2 \) → \( 4 < 2 \) (False)
- Check 2: \( 4 \geq -3(0) + 4 \) → \( 4 \geq 4 \) (True)
- Result: Not a solution (fails first inequality)
- Substitute \( x = 0 \) and \( y = 4 \):
-
For (4, 5):
- Substitute \( x = 4 \) and \( y = 5 \):
- Check 1: \( 5 < 2(4) + 2 \) → \( 5 < 10 \) (True)
- Check 2: \( 5 \geq -3(4) + 4 \) → \( 5 \geq -8 \) (True)
- Result: It is a solution (satisfies both inequalities)
- Substitute \( x = 4 \) and \( y = 5 \):
Summary of Results:
- (1, 4): Not a solution
- (2, 2): Solution
- (1, 1): Solution
- (0, 4): Not a solution
- (4, 5): Solution
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