Determine if the point (3,8) is a solution to the simultaneous inequalities y>x+5 and y≤6x−3 . Select the response that gives the correct answer as well as the correct reasoning.

Question

A: Yes, because both of these inequalities include points on the line as a solution

B: No, it is not in the solution set for either inequality

C: Yes it is within the solution set for both inequalities

D: No, it is in the solution set for y is less than or equal to 6x -3, but on the line for y > x + 5

Answer 1

To determine if the point (3,8) is a solution to the simultaneous inequalities \( y > x + 5 \) and \( y \leq 6x - 3 \), we will evaluate whether the point satisfies both inequalities.

1. Evaluate \( y > x + 5 \): \[ y = 8, \quad x = 3 \ 8 > 3 + 5 \ 8 > 8 \quad \text{(This is false since 8 is not greater than 8.)} \] Therefore, the point (3,8) does not satisfy \( y > x + 5 \).

2. Evaluate \( y \leq 6x - 3 \): \[ y = 8, \quad x = 3 \ 8 \leq 6 \cdot 3 - 3 \ 8 \leq 18 - 3 \ 8 \leq 15 \quad \text{(This is true.)} \] Therefore, the point (3,8) satisfies \( y \leq 6x - 3 \).

Since the point (3,8) does not satisfy the first inequality \( y > x + 5 \) but does satisfy the second inequality \( y \leq 6x - 3 \), the correct response is:

D: No, it is in the solution set for y ≤ 6x - 3, but on the line for y > x + 5.