Which of the following points shows a solution to the simultaneous inequalities y>12x+5 and y<−2x+1 ?(1 point)

Question

Responses

(4,8)
left parenthesis 4 comma 8 right parenthesis

(−1.6,4.2)
left parenthesis negative 1.6 comma 4.2 right parenthesis

(1,5.5)
left parenthesis 1 comma 5.5 right parenthesis

(−5,5)

Answer 1

To determine which of the given points satisfies the simultaneous inequalities \( y > 12x + 5 \) and \( y < -2x + 1 \), we need to evaluate each point against both inequalities.

1. Point (4, 8):

   • For \( y > 12x + 5 \): \( 8 > 12(4) + 5 \)
     \( 8 > 48 + 5 \)
     \( 8 > 53 \) (False)
   • For \( y < -2x + 1 \): \( 8 < -2(4) + 1 \)
     \( 8 < -8 + 1 \)
     \( 8 < -7 \) (False)

2. Point (-1.6, 4.2):

   • For \( y > 12x + 5 \): \( 4.2 > 12(-1.6) + 5 \)
     \( 4.2 > -19.2 + 5 \)
     \( 4.2 > -14.2 \) (True)
   • For \( y < -2x + 1 \): \( 4.2 < -2(-1.6) + 1 \)
     \( 4.2 < 3.2 + 1 \)
     \( 4.2 < 4.2 \) (False)

3. Point (1, 5.5):

   • For \( y > 12x + 5 \): \( 5.5 > 12(1) + 5 \)
     \( 5.5 > 12 + 5 \)
     \( 5.5 > 17 \) (False)
   • For \( y < -2x + 1 \): \( 5.5 < -2(1) + 1 \)
     \( 5.5 < -2 + 1 \)
     \( 5.5 < -1 \) (False)

4. Point (-5, 5):

   • For \( y > 12x + 5 \): \( 5 > 12(-5) + 5 \)
     \( 5 > -60 + 5 \)
     \( 5 > -55 \) (True)
   • For \( y < -2x + 1 \): \( 5 < -2(-5) + 1 \)
     \( 5 < 10 + 1 \)
     \( 5 < 11 \) (True)

Based on the evaluations, the only point that satisfies both inequalities is (-5, 5).