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 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 .

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.

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 can substitute \(x = 3\) and \(y = 8\) into each inequality.

1. Testing the first inequality \(y > x + 5\): \[ 8 > 3 + 5 \] \[ 8 > 8 \quad (\text{This is false.}) \]

2. Testing the second inequality \(y \leq 6x - 3\): \[ 8 \leq 6(3) - 3 \] \[ 8 \leq 18 - 3 \] \[ 8 \leq 15 \quad (\text{This is true.}) \]

Since the point (3, 8) does not satisfy the first inequality (it is not greater than \(x + 5\)), but it does satisfy the second inequality (it is less than or equal to \(6x - 3\)), the correct answer is:

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