Question

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

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

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

(1,5.5)
left parenthesis 1 comma 5.5 right parenthesis

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

Answer 1

To determine which of the given points satisfies the simultaneous inequalities \( y > \frac{1}{2} x + 5 \) and \( y < -2x + 1 \), we need to substitute each point into both inequalities.

1. **Point (-5, 5)**:
- For \( y > \frac{1}{2} x + 5 \):
- \( 5 > \frac{1}{2}(-5) + 5 \) ⇒ \( 5 > -2.5 + 5 \) ⇒ \( 5 > 2.5 \) (True)
- For \( y < -2x + 1 \):
- \( 5 < -2(-5) + 1 \) ⇒ \( 5 < 10 + 1 \) ⇒ \( 5 < 11 \) (True)
- Result: This point satisfies both inequalities.

2. **Point (4, 8)**:
- For \( y > \frac{1}{2} x + 5 \):
- \( 8 > \frac{1}{2}(4) + 5 \) ⇒ \( 8 > 2 + 5 \) ⇒ \( 8 > 7 \) (True)
- For \( y < -2x + 1 \):
- \( 8 < -2(4) + 1 \) ⇒ \( 8 < -8 + 1 \) ⇒ \( 8 < -7 \) (False)
- Result: This point does not satisfy the second inequality.

3. **Point (1, 5.5)**:
- For \( y > \frac{1}{2} x + 5 \):
- \( 5.5 > \frac{1}{2}(1) + 5 \) ⇒ \( 5.5 > 0.5 + 5 \) ⇒ \( 5.5 > 5.5 \) (False)
- Result: This point does not satisfy the first inequality.

4. **Point (-1.6, 4.2)**:
- For \( y > \frac{1}{2} x + 5 \):
- \( 4.2 > \frac{1}{2}(-1.6) + 5 \) ⇒ \( 4.2 > -0.8 + 5 \) ⇒ \( 4.2 > 4.2 \) (False)
- Result: This point does not satisfy the first inequality.

**Summary of Results:**
- Point (-5, 5) satisfies both inequalities.
- Points (4, 8), (1, 5.5), and (-1.6, 4.2) do not satisfy both inequalities.

Thus, the point that shows a solution to the simultaneous inequalities is **(-5, 5)**.