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

Question

Responses

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

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

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

(−5,5)

Answer 1

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

1. Point (1, 5.5):

   • Check \( y > 12x + 5 \): \[ 5.5 > 12(1) + 5 \implies 5.5 > 17 \quad \text{(False)} \]
   • Check \( y < -2x + 1 \): \[ 5.5 < -2(1) + 1 \implies 5.5 < -1 \quad \text{(False)} \]

2. Point (-1.6, 4.2):

   • Check \( y > 12x + 5 \): \[ 4.2 > 12(-1.6) + 5 \implies 4.2 > -19.2 + 5 \implies 4.2 > -14.2 \quad \text{(True)} \]
   • Check \( y < -2x + 1 \): \[ 4.2 < -2(-1.6) + 1 \implies 4.2 < 3.2 + 1 \implies 4.2 < 4.2 \quad \text{(False)} \]

3. Point (4, 8):

   • Check \( y > 12x + 5 \): \[ 8 > 12(4) + 5 \implies 8 > 48 + 5 \implies 8 > 53 \quad \text{(False)} \]
   • Check \( y < -2x + 1 \): \[ 8 < -2(4) + 1 \implies 8 < -8 + 1 \implies 8 < -7 \quad \text{(False)} \]

4. Point (-5, 5):

   • Check \( y > 12x + 5 \): \[ 5 > 12(-5) + 5 \implies 5 > -60 + 5 \implies 5 > -55 \quad \text{(True)} \]
   • Check \( y < -2x + 1 \): \[ 5 < -2(-5) + 1 \implies 5 < 10 + 1 \implies 5 < 11 \quad \text{(True)} \]

The only point that satisfies both inequalities is (-5, 5). Thus, the solution to the simultaneous inequalities is:

(-5, 5).