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

(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
Skip to navigation

Answer 1

To determine which point is a solution to the simultaneous inequalities \( y < \frac{1}{2}x \) and \( y > \frac{1}{3}x + 3 \), let's assess each option one by one.

1. For the inequality \( y < \frac{1}{2}x \):

   • This line has a slope of \(\frac{1}{2}\) and y-intercept 0. Points below this line satisfy the inequality.

2. For the inequality \( y > \frac{1}{3}x + 3 \):

   • This line has a slope of \(\frac{1}{3}\) and y-intercept 3. Points above this line satisfy the inequality.

Now, let's evaluate each point:

1. (25,12):

   • \( y < \frac{1}{2}(25) = 12.5 \) → \( 12 < 12.5 \) (True)
   • \( y > \frac{1}{3}(25) + 3 = \frac{25}{3} + 3 \approx 8.33 + 3 = 11.33 \) → \( 12 > 11.33 \) (True)
   • This point satisfies both inequalities.

2. (3,6):

   • \( y < \frac{1}{2}(3) = 1.5 \) → \( 6 < 1.5 \) (False)
   • This point does not satisfy the first inequality.

3. (2,2):

   • \( y < \frac{1}{2}(2) = 1 \) → \( 2 < 1 \) (False)
   • This point does not satisfy the first inequality.

4. (10,5):

   • \( y < \frac{1}{2}(10) = 5 \) → \( 5 < 5 \) (False)
   • This point does not satisfy the first inequality.

Given this analysis, the only point that satisfies both inequalities is:

(25,12)