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

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
−
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 .

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.

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 check each inequality individually.

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

   • Substitute \( x = 3 \) and \( y = 8 \): \[ 8 > 3 + 5 \] \[ 8 > 8 \]
   • This statement is false, as \( 8 \) is not greater than \( 8 \). Therefore, (3, 8) does not satisfy this inequality.

2. For the 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. Therefore, (3, 8) satisfies this inequality.

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

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