Determine if the point (3,8) is a solution to the simultaneous inequalities y>x+5 and y≤6x−3

Question

. Select the response that gives the correct answer as well as the correct reasoning.(1 point)
Responses

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 .
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.

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 with the point (3,8).

1. For the inequality \( y > x + 5 \): \[ 8 > 3 + 5 \ 8 > 8 \] This is not true, as 8 is not greater than 8.

2. For the inequality \( y \leq 6x - 3 \): \[ 8 \leq 6 \cdot 3 - 3 \ 8 \leq 18 - 3 \ 8 \leq 15 \] This is true, as 8 is indeed less than or equal to 15.

Since the point (3,8) does not satisfy the first inequality \( y > x + 5 \), but satisfies the second inequality \( y \leq 6x - 3 \), we can conclude that it is not a solution to the system of inequalities.

Now looking at the response options:

• "No, it is in the solution set for y≤6x−3, but on the line for y>x+5."

This option correctly states that the point satisfies one inequality and not the other.

Thus, the correct response is: No, it is in the solution set for y≤6x−3, but on the line for y>x+5.