Question

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

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

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

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

(1,5.5)

Answer 1

To determine which point satisfies the simultaneous inequalities \( y > 12x + 5 \) and \( y < -2x + 1 \), we'll evaluate each point against the inequalities.

1. **Point (4, 8)**:
- Check \( y > 12x + 5 \):
\[
8 > 12(4) + 5 \implies 8 > 48 + 5 \implies 8 > 53 \quad \text{(False)}
\]
- This point does not satisfy the first inequality.

2. **Point (-1.6, 4.2)**:
- Check \( y > 12x + 5 \):
\[
4.2 > 12(-1.6) + 5 \implies 4.2 > -19.2 + 5 \implies 4.2 > -14.2 \quad \text{(True)}
\]
- Check \( y < -2x + 1 \):
\[
4.2 < -2(-1.6) + 1 \implies 4.2 < 3.2 + 1 \implies 4.2 < 4.2 \quad \text{(False)}
\]
- This point does not satisfy the second inequality.

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

4. **Point (1, 5.5)**:
- Check \( y > 12x + 5 \):
\[
5.5 > 12(1) + 5 \implies 5.5 > 12 + 5 \implies 5.5 > 17 \quad \text{(False)}
\]
- This point does not satisfy the first inequality.

From this analysis, the only point that satisfies both inequalities is **(-5, 5)**.