Which of the following points shows a solution to the simultaneous inequalities y>1/2x+5

To find a solution to the simultaneous inequalities \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \), we need to check each of the provided points to see if they satisfy both inequalities.

1. Check (4, 8):

   • Inequality 1: \( 8 > \frac{1}{2}(4) + 5 \) → \( 8 > 2 + 5 \) → \( 8 > 7 \) (True)
   • Inequality 2: \( 8 < -2(4) + 1 \) → \( 8 < -8 + 1 \) → \( 8 < -7 \) (False)

2. Check (-1.6, 4.2):

   • Inequality 1: \( 4.2 > \frac{1}{2}(-1.6) + 5 \) → \( 4.2 > -0.8 + 5 \) → \( 4.2 > 4.2 \) (False, not greater)
   • Inequality 2: \( 4.2 < -2(-1.6) + 1 \) → \( 4.2 < 3.2 + 1 \) → \( 4.2 < 4.2 \) (False, not less)

3. Check (-5, 5):

   • Inequality 1: \( 5 > \frac{1}{2}(-5) + 5 \) → \( 5 > -2.5 + 5 \) → \( 5 > 2.5 \) (True)
   • Inequality 2: \( 5 < -2(-5) + 1 \) → \( 5 < 10 + 1 \) → \( 5 < 11 \) (True)

4. Check (1, 5.5):

   • Inequality 1: \( 5.5 > \frac{1}{2}(1) + 5 \) → \( 5.5 > 0.5 + 5 \) → \( 5.5 > 5.5 \) (False, not greater)
   • Inequality 2: \( 5.5 < -2(1) + 1 \) → \( 5.5 < -2 + 1 \) → \( 5.5 < -1 \) (False)

The only point that satisfies both inequalities is (-5, 5). Thus, the solution is:

(-5, 5).