Determine if the point (3,8) is a solution to the simultaneous inequalities y > x + 5 and y ≤ 62 - 3. Select the response that gives the correct answer as well as the correct reasoning.

Question

No, it is not in the solution set for either inequality.
Yes, because both of these inequalities include points on the line as a solution.
Yes, it is within the solution set for both inequalities.
No, it is in the solution set for y6z3, but on the line for y> +5.

Answer 1

To determine if the point (3, 8) is a solution to the simultaneous inequalities \( y > x + 5 \) and \( y \leq 62 - 3 \), we will evaluate each inequality separately.

1. Check the first inequality:
   \( y > x + 5 \)
   Substitute \( x = 3 \) and \( y = 8 \):
   \( 8 > 3 + 5 \)
   \( 8 > 8 \)
   This is false. (The inequality does not include the equal sign, so 8 is not greater than 8.)

2. Check the second inequality:
   \( y \leq 62 - 3 \)
   Simplify \( 62 - 3 \) to get 59. So the inequality becomes \( y \leq 59 \).
   Substitute \( y = 8 \):
   \( 8 \leq 59 \)
   This is true.

The point (3, 8) is not a solution to the first inequality (it lies on the line, but does not satisfy the strict inequality) but satisfies the second inequality.

Thus, the correct conclusion is: No, it is not in the solution set for either inequality.

The correct response is: No, it is in the solution set for \( y \leq 59 \), but on the line for \( y > x + 5 \).