Question

Which of the following points shows a solution to the simultaneous inequalities y>12x+5 and y<−2x+1 ?(1 point)
Responses

(−5,5)
left parenthesis negative 5 comma 5 right parenthesis

(−1.6,4.2)
left parenthesis negative 1.6 comma 4.2 right parenthesis

(4,8)
left parenthesis 4 comma 8 right parenthesis

(1,5.5)

Answer 1

To find which of the given points satisfies the simultaneous inequalities \( y > 12x + 5 \) and \( y < -2x + 1 \), we need to check each point against both inequalities.

1. **Point \( (-5, 5) \)**:
- For \( y > 12x + 5 \):
\[
5 > 12(-5) + 5 \implies 5 > -60 + 5 \implies 5 > -55 \quad (\text{True})
\]
- For \( y < -2x + 1 \):
\[
5 < -2(-5) + 1 \implies 5 < 10 + 1 \implies 5 < 11 \quad (\text{True})
\]
- **Result for point (-5, 5)**: Satisfies both inequalities.

2. **Point \( (-1.6, 4.2) \)**:
- For \( y > 12x + 5 \):
\[
4.2 > 12(-1.6) + 5 \implies 4.2 > -19.2 + 5 \implies 4.2 > -14.2 \quad (\text{True})
\]
- For \( y < -2x + 1 \):
\[
4.2 < -2(-1.6) + 1 \implies 4.2 < 3.2 + 1 \implies 4.2 < 4.2 \quad (\text{False})
\]
- **Result for point (-1.6, 4.2)**: Does not satisfy both inequalities.

3. **Point \( (4, 8) \)**:
- For \( y > 12x + 5 \):
\[
8 > 12(4) + 5 \implies 8 > 48 + 5 \implies 8 > 53 \quad (\text{False})
\]
- **Result for point (4, 8)**: Does not satisfy both inequalities.

4. **Point \( (1, 5.5) \)**:
- For \( y > 12x + 5 \):
\[
5.5 > 12(1) + 5 \implies 5.5 > 12 + 5 \implies 5.5 > 17 \quad (\text{False})
\]
- **Result for point (1, 5.5)**: Does not satisfy both inequalities.

Based on this analysis, the only point that satisfies both inequalities is \( (-5, 5) \).