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<1/2x and y>1/3x+3 ?

(1 point)
Responses

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

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

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

(10,5)

Answer 1

To determine which point is a solution to the simultaneous inequalities \( y < \frac{1}{2}x \) and \( y > \frac{1}{3}x + 3 \), we can analyze each point given in the options one by one by substituting the coordinates into the inequalities.

1. For the point (25, 12):

   • Check \( y < \frac{1}{2}x \):
     • \( 12 < \frac{1}{2}(25) = 12.5 \) → True
   • Check \( y > \frac{1}{3}x + 3 \):
     • \( 12 > \frac{1}{3}(25) + 3 \)
     • \( 12 > \frac{25}{3} + 3 \)
     • \( 12 > 8.33 + 3 \)
     • \( 12 > 11.33 \) → True
   • This point is a solution.

2. For the point (2, 2):

   • Check \( y < \frac{1}{2}x \):
     • \( 2 < \frac{1}{2}(2) = 1 \) → False
   • Not a solution.

3. For the point (3, 6):

   • Check \( y < \frac{1}{2}x \):
     • \( 6 < \frac{1}{2}(3) = 1.5 \) → False
   • Not a solution.

4. For the point (10, 5):

   • Check \( y < \frac{1}{2}x \):
     • \( 5 < \frac{1}{2}(10) = 5 \) → False (equal, so not less than)
   • Not a solution.

From the checks above, only the point (25, 12) satisfies both inequalities.

Final Answer: (25, 12)