Which ordered pair makes both inequalities true

Which ordered pair makes both inequalities true? y < 3x – 1 y > –x + 4
(4,0) (1,2) (0,4) (2,1)

The Correct Answer and Explanation is:

To solve the given system of inequalities, we need to check which ordered pair satisfies both inequalities:

1. ( y < 3x – 1 )
2. ( y > -x + 4 )

Step 1: Check each ordered pair

Let’s go through each ordered pair and check if it satisfies both inequalities.

Ordered Pair (4, 0):

Substitute ( x = 4 ) and ( y = 0 ) into both inequalities:

1. For ( y < 3x – 1 ):
   ( 0 < 3(4) – 1 )
   ( 0 < 12 – 1 )
   ( 0 < 11 )
   This is true.
2. For ( y > -x + 4 ):
   ( 0 > -(4) + 4 )
   ( 0 > -4 + 4 )
   ( 0 > 0 )
   This is false.

So, (4, 0) does not satisfy both inequalities.

Ordered Pair (1, 2):

Substitute ( x = 1 ) and ( y = 2 ) into both inequalities:

1. For ( y < 3x – 1 ):
   ( 2 < 3(1) – 1 )
   ( 2 < 3 – 1 )
   ( 2 < 2 )
   This is false.

Since the first inequality is not satisfied, we don’t need to check the second inequality. Therefore, (1, 2) does not satisfy both inequalities.

Ordered Pair (0, 4):

Substitute ( x = 0 ) and ( y = 4 ) into both inequalities:

1. For ( y < 3x – 1 ):
   ( 4 < 3(0) – 1 )
   ( 4 < 0 – 1 )
   ( 4 < -1 )
   This is false.

Since the first inequality is not satisfied, (0, 4) does not satisfy both inequalities.

Ordered Pair (2, 1):

Substitute ( x = 2 ) and ( y = 1 ) into both inequalities:

1. For ( y < 3x – 1 ):
   ( 1 < 3(2) – 1 )
   ( 1 < 6 – 1 )
   ( 1 < 5 )
   This is true.
2. For ( y > -x + 4 ):
   ( 1 > -(2) + 4 )
   ( 1 > -2 + 4 )
   ( 1 > 2 )
   This is false.

Thus, (2, 1) does not satisfy both inequalities.

Conclusion:

None of the ordered pairs satisfy both inequalities.