Question

Which point is a solution to the simultaneous inequalities y<12x and y>13x+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 \(y < 12x\) and \(y > 13x + 3\), we will evaluate each point against both inequalities.

1. **Point (25, 12)**:
- Check \( y < 12x \):
- \( 12 < 12 \times 25 \)?
- \( 12 < 300 \) (True)
- Check \( y > 13x + 3 \):
- \( 12 > 13 \times 25 + 3 \)?
- \( 12 > 325 + 3 \) → \( 12 > 328 \) (False)

2. **Point (2, 2)**:
- Check \( y < 12x \):
- \( 2 < 12 \times 2 \)?
- \( 2 < 24 \) (True)
- Check \( y > 13x + 3 \):
- \( 2 > 13 \times 2 + 3 \)?
- \( 2 > 26 + 3 \) → \( 2 > 29 \) (False)

3. **Point (3, 6)**:
- Check \( y < 12x \):
- \( 6 < 12 \times 3 \)?
- \( 6 < 36 \) (True)
- Check \( y > 13x + 3 \):
- \( 6 > 13 \times 3 + 3 \)?
- \( 6 > 39 + 3 \) → \( 6 > 42 \) (False)

4. **Point (10, 5)**:
- Check \( y < 12x \):
- \( 5 < 12 \times 10 \)?
- \( 5 < 120 \) (True)
- Check \( y > 13x + 3 \):
- \( 5 > 13 \times 10 + 3 \)?
- \( 5 > 130 + 3 \) → \( 5 > 133 \) (False)

None of the points \((25, 12)\), \((2, 2)\), \((3, 6)\), or \((10, 5)\) satisfy both inequalities simultaneously.

Based on the results above:

- All points satisfy the first inequality \(y < 12x\), but none satisfy the second inequality \(y > 13x + 3\).

Thus, there are **no solutions** among the given points to both inequalities.