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

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

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.

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

   • Substitute \( x = 3 \) and \( y = 8 \): \[ 8 > 3 + 5 \implies 8 > 8 \] This inequality is not true since 8 is not greater than 8. Therefore, (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 inequality is true since 8 is less than or equal to 15. Therefore, (3, 8) satisfies this inequality.

Now summarizing the results:

• The point (3, 8) is not a solution for the inequality \( y > x + 5 \).
• The point (3, 8) is a solution for the inequality \( y \leq 6x - 3 \).

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