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

(25,12)
left parenthesis 25 comma 12 right parenthesis

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

(10,5)

Answer 1

To determine which point is a solution to the simultaneous inequalities \( y < 12x \) and \( y > 13x + 3 \), we need to check each point against both inequalities.

1. For the point \( (3, 6) \):

   • Check \( y < 12x \): \( 6 < 12(3) \) → \( 6 < 36 \) (True)
   • Check \( y > 13x + 3 \): \( 6 > 13(3) + 3 \) → \( 6 > 39 + 3 \) → \( 6 > 42 \) (False)

2. For the point \( (25, 12) \):

   • Check \( y < 12x \): \( 12 < 12(25) \) → \( 12 < 300 \) (True)
   • Check \( y > 13x + 3 \): \( 12 > 13(25) + 3 \) → \( 12 > 325 + 3 \) → \( 12 > 328 \) (False)

3. For the point \( (2, 2) \):

   • Check \( y < 12x \): \( 2 < 12(2) \) → \( 2 < 24 \) (True)
   • Check \( y > 13x + 3 \): \( 2 > 13(2) + 3 \) → \( 2 > 26 + 3 \) → \( 2 > 29 \) (False)

4. For the point \( (10, 5) \):

   • Check \( y < 12x \): \( 5 < 12(10) \) → \( 5 < 120 \) (True)
   • Check \( y > 13x + 3 \): \( 5 > 13(10) + 3 \) → \( 5 > 130 + 3 \) → \( 5 > 133 \) (False)

None of the points satisfy both inequalities. Therefore, based on these checks, it appears that there is no solution among the provided points that satisfies both inequalities \( y < 12x \) and \( y > 13x + 3 \).