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 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
, 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 .

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.

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 will evaluate each inequality separately.

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; it is equal to 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** because 8 is less than or equal to 15.

Now, combining the results:
- The point (3, 8) satisfies the second inequality \( y \leq 6x - 3 \) but does not satisfy the first inequality \( y > x + 5 \).

Based on this evaluation:

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