Determine if the point (3,8)

Question

(
3
,
8
)
is a solution to the simultaneous inequalities y>x+5
𝑦
>
𝑥
+
5
and y≤6x−3
𝑦
≤
6
𝑥
−
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
𝑦
≤
6
𝑥
−
3
, but on the line for y>x+5
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.

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 not in the solution set for either inequality.

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 with the point coordinates \( x = 3 \) and \( y = 8 \).

1. Check the first inequality:
   \( y > x + 5 \)
   Substitute \( x = 3 \) and \( y = 8 \):
   \( 8 > 3 + 5 \)
   \( 8 > 8 \)
   This is false because 8 is not greater than 8.

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

Since the point (3, 8) does not satisfy the first inequality (it is not greater), but satisfies the second inequality, we conclude:

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