Question

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) 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 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 can evaluate each inequality separately with the point \( (3, 8) \).

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

   • Substitute \( x = 3 \) and \( y = 8 \): \[ 8 > 3 + 5 \implies 8 > 8 \]
   • This is false, as \( 8 \) is not greater than \( 8 \); instead, it is equal.

2. Check 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 is true.

Now we've determined:

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

In conclusion, since \( (3, 8) \) is not a solution to both inequalities, the correct response is:

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