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

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

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, 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 will evaluate each inequality with this point.

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

Plug in \( x = 3 \) and \( y = 8 \):
\[
8 > 3 + 5
\]
\[
8 > 8
\]
This statement is **false** because 8 is not greater than 8. Thus, the point (3, 8) does not satisfy this inequality.

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

Again, plug in \( x = 3 \) and \( y = 8 \):
\[
8 \leq 6(3) - 3
\]
\[
8 \leq 18 - 3
\]
\[
8 \leq 15
\]
This statement is **true** because 8 is indeed less than or equal to 15.

Since the point (3, 8) satisfies the second inequality but does not satisfy the first inequality, the appropriate response is:

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