Question

Determine if the point (3,8)
(
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

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.
No, it is not in the solution set for either inequality.

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
𝑦
≤
6
𝑥
−

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 test the point against both inequalities.

1. **Test the first inequality \( y > x + 5 \)**:
- Substitute \( x = 3 \) and \( y = 8 \):
\[
8 > 3 + 5
\]
\[
8 > 8
\]
- This statement is false because \( 8 \) is not greater than \( 8 \). Therefore, the point (3, 8) does not satisfy the first inequality.

2. **Test 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 statement is true. The point (3, 8) satisfies the second inequality.

Since the point (3, 8) does not satisfy the first inequality but satisfies the second, it is not a solution to the simultaneous inequalities.

Therefore, the correct response is:
**No, it is not in the solution set for either inequality.**