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.(1 point)

Question

Responses

No, it is not in the solution set for either inequality.
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, because both of these inequalities include points on the line as a solution.

Yes, it is within the solution set for both inequalities.
Yes, it is within the solution set for both inequalities.

No, it is in the solution set for y≤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 need to evaluate each inequality separately.

1. For the first inequality \( y > x + 5 \):

   • Substitute \( x = 3 \) and \( y = 8 \): \[ 8 > 3 + 5 \] This simplifies to: \[ 8 > 8 \] This is false. The point (3,8) does not satisfy this inequality.

2. For the second inequality \( y \leq 6x - 3 \):

   • Substitute \( x = 3 \) and \( y = 8 \): \[ 8 \leq 6(3) - 3 \] This simplifies to: \[ 8 \leq 18 - 3 \] \[ 8 \leq 15 \] This is true. The point (3,8) satisfies this inequality.

Since the point (3,8) does not satisfy the first inequality but does satisfy the second inequality, we can conclude that it is not a solution to both inequalities simultaneously.

The correct response is: No, it is in the solution set for \( y \leq 6x - 3 \), but on the line for \( y > x + 5 \).