Question

Systems of linear inequalities solutions
Determine if each ordered pair is a solution of the system of linear inequalities.
#1 y<2x+2 & y≥-3x+4
(1,4) (2,2) (1,1) (0,4) (4,5)

Answer 1

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:
1. \( y < 2x + 2 \)
2. \( y \geq -3x + 4 \)

### Check each ordered pair:

1. **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)

2. **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)

3. **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)

4. **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)

5. **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)

### Summary of Results:
- **(1, 4)**: Not a solution
- **(2, 2)**: Solution
- **(1, 1)**: Solution
- **(0, 4)**: Not a solution
- **(4, 5)**: Solution