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, 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 simultaneous inequalities \( y > x + 5 \) and \( y \leq 6x - 3 \), we need to check whether the point satisfies each inequality.

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

   • Substitute \( x = 3 \) into the inequality: \[ y > 3 + 5 \implies y > 8 \]
   • The point (3,8) gives \( y = 8 \), which does not satisfy \( y > 8 \). Therefore, this inequality is not satisfied.

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

   • Substitute \( x = 3 \) into the inequality: \[ y \leq 6(3) - 3 \implies y \leq 18 - 3 \implies y \leq 15 \]
   • The point (3,8) gives \( y = 8 \), which satisfies \( y \leq 15 \).

Now, summarizing the results:

• The point (3,8) does not satisfy the first inequality \( y > x + 5 \).
• The point (3,8) does satisfy the second inequality \( y \leq 6x - 3 \).

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