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 determine which of the given points satisfies the simultaneous inequalities \( y > 12x + 5 \) and \( y < -2x + 1 \), we will evaluate each point one by one against both inequalities.

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

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

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

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

Final Conclusion: The only point that satisfies both inequalities \( y > 12x + 5 \) and \( y < -2x + 1 \) is \((-5, 5)\).