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, 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 .

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.

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 if the point satisfies both inequalities.

1. Check the first inequality \( y > x + 5 \): Plugging in \( x = 3 \) and \( y = 8 \): \[ 8 > 3 + 5 \] \[ 8 > 8 \] This is false since 8 is not greater than 8. Therefore, (3, 8) does not satisfy this inequality.

2. Check the second inequality \( y \leq 6x - 3 \): Plugging in \( x = 3 \) and \( y = 8 \): \[ 8 \leq 6(3) - 3 \] \[ 8 \leq 18 - 3 \] \[ 8 \leq 15 \] This is true.

Since (3, 8) satisfies the inequality \( y \leq 6x - 3 \) but does not satisfy the inequality \( y > x + 5 \), we conclude that:

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