Question

Use the image to answer the 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

(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 \( y < \frac{1}{2}x \) and \( y > \frac{1}{3}x + 3 \), we need to analyze each inequality:

1. **First Inequality: \( y < \frac{1}{2}x \)**
- This inequality represents the region below the line \( y = \frac{1}{2}x \).

2. **Second Inequality: \( y > \frac{1}{3}x + 3 \)**
- This inequality represents the region above the line \( y = \frac{1}{3}x + 3 \).

Next, we will plug in each of the given points to see if they satisfy both inequalities.

1. **Point (3, 6)**:
- For \( y < \frac{1}{2}x \): \( 6 < \frac{1}{2} \cdot 3 \) → \( 6 < 1.5 \) (false)
- For \( y > \frac{1}{3}x + 3\): \( 6 > \frac{1}{3} \cdot 3 + 3 \) → \( 6 > 4 \) (true)

This point does not satisfy the first inequality.

2. **Point (25, 12)**:
- For \( y < \frac{1}{2}x \): \( 12 < \frac{1}{2} \cdot 25 \) → \( 12 < 12.5 \) (true)
- For \( y > \frac{1}{3}x + 3 \): \( 12 > \frac{1}{3} \cdot 25 + 3 \) → \( 12 > \frac{25}{3} + 3 \) → \( 12 > 8.33 + 3 \) → \( 12 > 11.33 \) (true)

This point satisfies both inequalities.

3. **Point (2, 2)**:
- For \( y < \frac{1}{2}x \): \( 2 < \frac{1}{2} \cdot 2 \) → \( 2 < 1 \) (false)
- For \( y > \frac{1}{3}x + 3 \): \( 2 > \frac{1}{3} \cdot 2 + 3 \) → \( 2 > 0.67 + 3 \) → \( 2 > 3.67 \) (false)

This point does not satisfy either inequality.

4. **Point (10, 5)**:
- For \( y < \frac{1}{2}x \): \( 5 < \frac{1}{2} \cdot 10 \) → \( 5 < 5 \) (false)
- For \( y > \frac{1}{3}x + 3 \): \( 5 > \frac{1}{3} \cdot 10 + 3 \) → \( 5 > 3.33 + 3 \) → \( 5 > 6.33 \) (false)

This point does not satisfy either inequality.

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

**(25, 12)**

Therefore, the answer is:
**(25,12)**