Systems of linear inequalities solutions

Question

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 $y\ge -3x+4$, we will substitute each ordered pair into both inequalities and check if they satisfy both conditions.

Inequalities:

1. $y<2x+2$
2. $y\ge -3x+4$

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: $4\ge -3(1)+4$ → $4\ge 1$ (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: $2\ge -3(2)+4$ → $2\ge -2$ (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: $1\ge -3(1)+4$ → $1\ge 1$ (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: $4\ge -3(0)+4$ → $4\ge 4$ (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: $5\ge -3(4)+4$ → $5\ge -8$ (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