Use the image to answer the question.

Question

A Cartesian coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. 2 dashed lines with arrows at both ends are plotted, and various regions of the graph are marked.

Which point is a solution to the simultaneous inequalities y<12x
and y>13x+3
?

(1 point)
Responses

(10,5)
left parenthesis 10 comma 5 right parenthesis

(3,6)
left parenthesis 3 comma 6 right parenthesis

(2,2)
left parenthesis 2 comma 2 right parenthesis

(25,12)

Answer 1

To determine which point is a solution to the simultaneous inequalities \(y < 12x\) and \(y > 13x + 3\), we can evaluate each point one by one.

1. For the point (10, 5):

   • Check \(y < 12x\): \[ 5 < 12(10) \implies 5 < 120 \quad \text{(True)} \]
   • Check \(y > 13x + 3\): \[ 5 > 13(10) + 3 \implies 5 > 130 + 3 \implies 5 > 133 \quad \text{(False)} \]

2. For the point (3, 6):

   • Check \(y < 12x\): \[ 6 < 12(3) \implies 6 < 36 \quad \text{(True)} \]
   • Check \(y > 13x + 3\): \[ 6 > 13(3) + 3 \implies 6 > 39 + 3 \implies 6 > 42 \quad \text{(False)} \]

3. For the point (2, 2):

   • Check \(y < 12x\): \[ 2 < 12(2) \implies 2 < 24 \quad \text{(True)} \]
   • Check \(y > 13x + 3\): \[ 2 > 13(2) + 3 \implies 2 > 26 + 3 \implies 2 > 29 \quad \text{(False)} \]

4. For the point (25, 12):

   • Check \(y < 12x\): \[ 12 < 12(25) \implies 12 < 300 \quad \text{(True)} \]
   • Check \(y > 13x + 3\): \[ 12 > 13(25) + 3 \implies 12 > 325 + 3 \implies 12 > 328 \quad \text{(False)} \]

None of the points satisfy both inequalities simultaneously. Therefore, none of the given points are solutions to the inequalities \(y < 12x\) and \(y > 13x + 3\).

If you need to proceed with different points or have other details that may affect the inequalities, please provide further context!