Solution Sets of Simultaneous Inequalities Quick Check

Question

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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 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 .

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.

Yes, it is within the solution set for both inequalities.
Yes, it is within the solution set for both inequalities.
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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 substitute \(x = 3\) and \(y = 8\) into each inequality.

1. First Inequality: \(y > x + 5\)

   Substituting the point (3, 8): \[ 8 > 3 + 5 \] \[ 8 > 8 \] This inequality is not satisfied because 8 is not greater than 8; it is equal.

2. Second Inequality: \(y \leq 6x - 3\)

   Substituting the point (3, 8): \[ 8 \leq 6(3) - 3 \] \[ 8 \leq 18 - 3 \] \[ 8 \leq 15 \] This inequality is satisfied since 8 is less than or equal to 15.

Since the point (3, 8) does not satisfy the first inequality \(y > x + 5\), it is not in the solution set for this inequality, even though it is in the solution set for the second inequality \(y \leq 6x - 3\).

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