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 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

Answer 1

To determine whether the point (3, 8) is a solution to the given inequalities, we can substitute the x and y coordinates into each inequality.

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

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

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

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

Based on the evaluations above, the point (3, 8) does not satisfy the first inequality \( y > x + 5 \) (it is actually on the line), but it satisfies the second inequality \( y \leq 6x - 3 \).

Therefore, the correct response is:

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