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<12x and y>13x+3 ?

(1 point)
Responses

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

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

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

(25,12)
left parenthesis 25 comma 12 right parenthesis
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Answer 1

To determine which point is a solution to the inequalities \( y < 12x \) and \( y > 13x + 3 \), we will evaluate each of the given points against both inequalities.

1. **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)

2. **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)

3. **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)

4. **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)

Since none of the points satisfy both inequalities \( y < 12x \) and \( y > 13x + 3 \), it appears there is no solution among the provided options. If you need further assistance or clarification, feel free to ask!