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

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.

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

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.
No, it is in the solution set for y less-than-or-equal-to 6 x minus 3 , but on the line for y greater than x plus 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 check each inequality:

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

   • Substitute \( x = 3 \) and \( y = 8 \): \[ 8 > 3 + 5 \implies 8 > 8 \] This is not true because 8 is not greater than 8.

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

   • Substitute \( x = 3 \) and \( y = 8 \): \[ 8 \leq 6(3) - 3 \implies 8 \leq 18 - 3 \implies 8 \leq 15 \] This is true because 8 is less than or equal to 15.

Based on this analysis:

• The point (3, 8) is not a solution for the first inequality \( y > x + 5 \).
• The point (3, 8) is a solution for the second inequality \( y \leq 6x - 3 \).

Therefore, the correct response is:

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