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

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

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

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

(25,12)

Answer 1

To determine which point is a solution to the simultaneous inequalities \( y < 12x \) and \( y > 13x + 3 \), let's analyze the inequalities one by one.

1. For the first inequality \( y < 12x \):

   • This describes the region below the line \( y = 12x \).

2. For the second inequality \( y > 13x + 3 \):

   • This describes the region above the line \( y = 13x + 3 \).

To find if a point is a solution for both inequalities, we will check each option:

1. Point (3, 6):

   • \( y = 6 \) and \( 12x = 12(3) = 36 \) → \( 6 < 36 \) (true)
   • \( y = 6 \) and \( 13x + 3 = 13(3) + 3 = 39 + 3 = 42 \) → \( 6 > 42 \) (false)
   • Not a solution.

2. Point (2, 2):

   • \( y = 2 \) and \( 12x = 12(2) = 24 \) → \( 2 < 24 \) (true)
   • \( y = 2 \) and \( 13x + 3 = 13(2) + 3 = 26 + 3 = 29 \) → \( 2 > 29 \) (false)
   • Not a solution.

3. Point (10, 5):

   • \( y = 5 \) and \( 12x = 12(10) = 120 \) → \( 5 < 120 \) (true)
   • \( y = 5 \) and \( 13x + 3 = 13(10) + 3 = 130 + 3 = 133 \) → \( 5 > 133 \) (false)
   • Not a solution.

4. Point (25, 12):

   • \( y = 12 \) and \( 12x = 12(25) = 300 \) → \( 12 < 300 \) (true)
   • \( y = 12 \) and \( 13x + 3 = 13(25) + 3 = 325 + 3 = 328 \) → \( 12 > 328 \) (false)
   • Not a solution.

All points tested do not satisfy both inequalities. Please check the selected points against the inequalities or provide a graph/visual to re-evaluate the area defined by the inequalities.