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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 .

Yes, it is within the solution set for both inequalities.
Yes, it is within the solution set for both inequalities.

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.

5 answers

To determine if the point (3, 8) is a solution to the simultaneous inequalities \( y > x + 5 \) and \( y \leq 6x - 3 \), we will check each inequality separately:

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

    Plugging in \( x = 3 \) and \( y = 8 \): \[ 8 > 3 + 5 \ 8 > 8 \] This is false because 8 is not greater than 8.

  2. For the inequality \( y \leq 6x - 3 \):

    Plugging in \( x = 3 \) and \( y = 8 \): \[ 8 \leq 6(3) - 3 \ 8 \leq 18 - 3 \ 8 \leq 15 \] This is true.

Now we summarize the results:

  • The point (3, 8) does not satisfy \( y > x + 5 \) (it lies on the line).
  • The point (3, 8) does satisfy \( y \leq 6x - 3 \).

Based on this reasoning, the correct response is:

No, it is in the solution set for y≤6x−3, but on the line for y>x+5.

Use the image to answer the question.

Which point is a solution to the simultaneous inequalities y<12x and y>13x+3?

(1 point)
Responses

(2,2)
left parenthesis 2 comma 2 right parenthesis

(3,6)
left parenthesis 3 comma 6 right parenthesis

(25,12)
left parenthesis 25 comma 12 right parenthesis

(10,5)

To determine which point is a solution to the simultaneous inequalities \( y < 12x \) and \( y > 13x + 3 \), we can analyze each point given in the responses.

We need to check if each point satisfies both inequalities:

  1. Point (2, 2):

    • Check \( y < 12x \):
      \( 2 < 12(2) \) → \( 2 < 24 \) (True)
    • Check \( y > 13x + 3 \):
      \( 2 > 13(2) + 3 \) → \( 2 > 26 + 3 \) → \( 2 > 29 \) (False)

  2. Point (3, 6):

    • Check \( y < 12x \):
      \( 6 < 12(3) \) → \( 6 < 36 \) (True)
    • Check \( y > 13x + 3 \):
      \( 6 > 13(3) + 3 \) → \( 6 > 39 + 3 \) → \( 6 > 42 \) (False)

  3. Point (25, 12):

    • Check \( y < 12x \):
      \( 12 < 12(25) \) → \( 12 < 300 \) (True)
    • Check \( y > 13x + 3 \):
      \( 12 > 13(25) + 3 \) → \( 12 > 325 + 3 \) → \( 12 > 328 \) (False)

  4. Point (10, 5):

    • Check \( y < 12x \):
      \( 5 < 12(10) \) → \( 5 < 120 \) (True)
    • Check \( y > 13x + 3 \):
      \( 5 > 13(10) + 3 \) → \( 5 > 130 + 3 \) → \( 5 > 133 \) (False)

After checking all points, we find that none of the points satisfy both inequalities.

However, if you're asking for the closest point that satisfies at least one inequality:

  • All four points satisfy \( y < 12x \), but none of them satisfy \( y > 13x + 3 \).

Therefore, there is no single point that is a solution to both inequalities based on the checks. If there are specific boundary conditions or constraints, more context would help clarify the situation.

you have to choose one

I apologize for the confusion in my previous analysis. Since we need to find one point that satisfies both inequalities \( y < 12x \) and \( y > 13x + 3 \), let's re-evaluate the options with a focus on one inequality at a time.

Given Points:

  1. (2, 2)
  2. (3, 6)
  3. (25, 12)
  4. (10, 5)

Checking Each Point

  1. Point (2, 2):

    • \( y < 12x \):
      • \( 2 < 12(2) \) → \( 2 < 24 \) (True)
    • \( y > 13x + 3 \):
      • \( 2 > 13(2) + 3 \) → \( 2 > 26 + 3 \) → \( 2 > 29 \) (False)

  2. Point (3, 6):

    • \( y < 12x \):
      • \( 6 < 12(3) \) → \( 6 < 36 \) (True)
    • \( y > 13x + 3 \):
      • \( 6 > 13(3) + 3 \) → \( 6 > 39 + 3 \) → \( 6 > 42 \) (False)

  3. Point (25, 12):

    • \( y < 12x \):
      • \( 12 < 12(25) \) → \( 12 < 300 \) (True)
    • \( y > 13x + 3 \):
      • \( 12 > 13(25) + 3 \) → \( 12 > 325 + 3 \) → \( 12 > 328 \) (False)

  4. Point (10, 5):

    • \( y < 12x \):
      • \( 5 < 12(10) \) → \( 5 < 120 \) (True)
    • \( y > 13x + 3 \):
      • \( 5 > 13(10) + 3 \) → \( 5 > 130 + 3 \) → \( 5 > 133 \) (False)

Conclusion

None of the points satisfy both inequalities. However, since the task requires us to choose one point, we should select the one that satisfies the most conditions.

All points satisfy \( y < 12x \), which could indicate they lie under that boundary line, but none satisfy \( y > 13x + 3 \). In this case, if I had to choose one, I would recommend going with one of the points that satisfies the \( y < 12x \) condition. The best choice would be point (10, 5) since it has the least gap from the boundary line defined by \( y = 12x \).

Final Answer:

(10, 5)