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.

Question

A. Yes, it is within the solution set for both inequalities
B. No, it is in the solution set for y < 6x – 3, but on the line for y > x + 5.
C. No, it is not in the solution set for either inequality.
D. 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 inequalities \( y > x + 5 \) and \( y < 6x - 3 \), we will evaluate each inequality at the point (3, 8).

1. For the inequality \( y > x + 5 \): \[ y > x + 5 \implies 8 > 3 + 5 \implies 8 > 8 \] This statement is false because 8 is not greater than 8. Therefore, (3, 8) does not satisfy the first inequality.

2. For the inequality \( y < 6x - 3 \): \[ y < 6x - 3 \implies 8 < 6(3) - 3 \implies 8 < 18 - 3 \implies 8 < 15 \] This statement is true.

Now, we can summarize the findings:

• (3, 8) does not satisfy \( y > x + 5 \).
• (3, 8) satisfies \( y < 6x - 3 \).

Now we can select the correct response based on this reasoning:

• A is incorrect because (3, 8) does not satisfy both inequalities.
• B is correct because (3, 8) is not in the solution set for \( y > x + 5 \) (it's on the line) and is in the solution set for \( y < 6x - 3 \).
• C is incorrect because (3, 8) does satisfy one of the inequalities.
• D is incorrect because the inequalities do not include points on the lines as solutions.

Thus, the correct answer is: B. No, it is in the solution set for \( y < 6x - 3 \), but on the line for \( y > x + 5 \).