Determine if the point (3,8)

Question

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)
Responses

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

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.

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 inequalities \( y > x + 5 \) and \( y \leq 6x - 3 \), we will evaluate both inequalities separately.

1. First Inequality: \( y > x + 5 \)

   For the point (3, 8): \[ 8 > 3 + 5 \] Simplifying the right side: \[ 8 > 8 \] This is false because 8 is not greater than 8.

2. Second Inequality: \( y \leq 6x - 3 \)

   For the point (3, 8): \[ 8 \leq 6(3) - 3 \] Simplifying the right side: \[ 8 \leq 18 - 3 \] \[ 8 \leq 15 \] This is true because 8 is less than or equal to 15.

Conclusion: The point (3, 8) satisfies the second inequality but does not satisfy the first inequality. Therefore, the correct response is:

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