gistfile1.txt

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[system]
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<Instructions>
Find a solution to the provided mathematical problem. The answer is a unique mathematical expression presented in LaTeX `\boxed{}` directive.  (example: `\boxed{4}` or `\boxed{3\pi}`). Formatting instructions: fractions should be represented in the LaTeX form `\frac{a}{b}` (not `\frac12`), units should not be included, square roots should be presented in the LaTeX form `\sqrt{c}` (not `\sqrt2`), all spaces and non critical parentheses or formatting should be stripped, rational numbers should be presented with a leading `0`.

Provide a reasoning consisting in multiple steps, using one line per step. A reasoning step is one coherent step of mathematical reasoning. It should hold in one line of at most 500 characters. If an answer is reached as part of the reasoning, it should be included in the reasoning step using the `\boxed{}` directive. Don't use the `\boxed{}` directive for anything else than the answer.
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You are an expert professor in your field of expertise. A good explanation is minimal, deductive, correct and complete. It should be clearly understandable by your PhD students, ommiting obvious details but including all the necessary steps to reach the conclusion. Be precise about what you think is good or bad in the proposed explanation. Try to think hard about what might be incorrect in the explanation and always propose ways to improve it to make it clearer, more concise if possible, more precise if necessary, and more convincing.

Based on the following 4 explanations from field experts along with commentaries made by other experts on each of them, your goal is to propose the best possible explanation to answer the following question:

QUESTION: For what values of $x$ is it true that $x^2 - 5x - 4 \le 10$? Express your answer in interval notation.
</Instructions>

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[user]
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The explanations and commentaries:

EXPLANATION 0:

REASONING:
We start by solving the quadratic equation $x^2-5x-4=10$
This simplifies to $x^2-5x-14=0$
Factoring, we get $(x-7)(x+2)=0$
So the solutions are $x=7$ and $x=-2$
These values divide the real number line into three intervals: $(-\infty,-2)$, $(-2,7)$, and $(7,\infty)$
We can test a value in each interval to see if it satisfies the inequality
For $x=-3$, we have $(-3)^2-5(-3)-4=9+15-4=20$, which is not less than or equal to 10
For $x=0$, we have $0^2-5(0)-4=-4$, which is less than or equal to 10
For $x=8$, we have $8^2-5(8)-4=64-40-4=20$, which is not less than or equal to 10
So the solution is the interval where the inequality holds, which is $\boxed{(-2,7)}$
ANSWER: \boxed{(-2,7)}

EXPERT COMMENTARY 0 0:

The explanation provided is correct and clear in its approach to solving the inequality. The steps are well laid out and the reasoning is sound.

One possible improvement would be to mention that the quadratic equation $x^2-5x-4=10$ is first rearranged to $x^2-5x-14=0$ before factoring. This would make the process more understandable for students who might not be familiar with directly solving the equation in its given form.

The method of testing a value in each interval to determine the solution is a valid and systematic approach. It's good that the explanation includes this step as it provides a way to confirm the solution.

Overall, the solution is correctly obtained and the reasoning is well-explained. The use of interval notation to express the final answer is also appropriate and clear.

The explanation is clear, concise, and logically sound. It effectively guides the reader through the process of solving the inequality.

EXPERT COMMENTARY 0 1:

The expert's commentary is mostly accurate and provides a positive assessment of the given explanation. However, there are a few points that need to be addressed to improve the clarity and correctness of the explanation.

1. The expert correctly points out that rearranging the given quadratic equation $x^2-5x-4=10$ to $x^2-5x-14=0$ before factoring would make the process more understandable. This is a good suggestion for clarity, especially for students who might be less familiar with solving quadratic equations directly.

2. The method of testing a value in each interval to determine the solution is indeed a valid and systematic approach. However, it would be beneficial to explain why this method works. By mentioning that the inequality $x^2-5x-4 \le 10$ represents a parabola opening upwards, and testing points in each interval helps determine where the parabola is below or at the level of $10$, the explanation would be more complete.

3. It would be helpful to add a concluding statement explicitly stating that the solution is the interval where the inequality holds, which is $(-2,7)$, and then express this in interval notation.

Overall, the given explanation is clear, logically sound, and effectively guides the reader through the process of solving the inequality. With the suggested improvements, the explanation would be more comprehensive and precise.

EXPERT COMMENTARY 0 2:

The expert's commentary provides a positive assessment of the given explanation, acknowledging its correctness and clarity. However, there are some critical points that need to be addressed to improve the precision and completeness of the explanation.

1. The suggestion to rearrange the quadratic equation $x^2-5x-4=10$ to $x^2-5x-14=0$ before factoring is crucial for clarity. This step should be explicitly included in the reasoning to ensure a complete and understandable solution.

2. While the method of testing a value in each interval to determine the solution is valid, it would be beneficial to explain why this method works. Mentioning that the inequality $x^2-5x-4 \le 10$ represents a parabola opening upwards and testing points in each interval helps determine where the parabola is below or at the level of $10$ would provide a more complete explanation.

3. The conclusion should explicitly state that the solution is the interval where the inequality holds, which is $(-2,7)$, and then express this in interval notation. This final step is essential for a complete and precise solution.

In summary, the given explanation is clear and logically sound, but it would benefit from including the suggested improvements to make it more comprehensive and precise.

EXPLANATION 1:

REASONING:
First, move the constant to the right side to get $x^2 - 5x - 14 \le 0$
Next, factor the quadratic to get $(x-7)(x+2) \le 0$
The solutions to this inequality are the values of $x$ for which the quadratic expression is less than or equal to zero
The solutions are $-2 \le x \le 7$
Expressing this in interval notation gives $\boxed{[-2, 7]}$
ANSWER: \boxed{[-2, 7]}

EXPERT COMMENTARY 1 0:

The initial steps to move the constant and factor the quadratic are correct.
To find the solutions, we need to consider the signs of the factors $(x-7)$ and $(x+2)$.
The inequality $(x-7)(x+2) \le 0$ is true when one factor is positive and the other is negative, or when the whole expression is zero.
This occurs for $-2 \le x \le 7$, which includes the boundary points.
Expressing this in interval notation gives the correct answer $\boxed{[-2, 7]}$.

The explanation is clear and correct. It logically and accurately guides the reader through the steps to find the solution. There are no unnecessary details, and the reasoning is concise and complete. This is a good explanation that effectively communicates the solution to the problem.

EXPERT COMMENTARY 1 1:

The explanation provided is clear and correct. The reasoning is logically sound and leads to the correct conclusion. The steps taken to move the constant and factor the quadratic are accurate. The consideration of the signs of the factors $(x-7)$ and $(x+2)$ is well explained, and the conclusion that the inequality $(x-7)(x+2) \le 0$ is true when one factor is positive and the other is negative, or when the whole expression is zero, is accurate.

The interval notation $[-2, 7]$ is correctly obtained as the solution to the inequality. The explanation is concise and complete, providing a clear path to the answer.

Overall, the given explanation is a good example of how to solve the inequality and express the solution in interval notation. There are no unnecessary details, and the reasoning is presented clearly and coherently.

Therefore, the commentary provided by Expert 0 is accurate and does not require any criticism.

EXPERT COMMENTARY 1 2:

Expert 0 and Expert 1 provided positive feedback on the explanation, but there are some points that need to be addressed:

1. The explanation should explicitly mention the method of finding the solutions by considering the signs of the factors and the use of the zero product property.

2. It is important to emphasize that the boundary points should be included in the solution set when expressing the solution in interval notation.

3. While the reasoning is clear, it could be improved by explicitly stating the method used to find the solution, such as using the factored form of the quadratic and considering the signs of the factors.

4. The explanation could benefit from a brief mention of the zero product property, which is the key concept behind solving the inequality using the factored form.

In summary, the given explanation is correct, but it could be improved by explicitly stating the method used to find the solution and emphasizing the inclusion of boundary points when expressing the solution in interval notation.

EXPLANATION 2:

REASONING:
We start by solving the quadratic equation $x^2-5x-4=10$
This simplifies to $x^2-5x-14=0$
Factoring, we get $(x-7)(x+2)=0$
So the solutions are $x=7$ and $x=-2$
These values divide the real number line into three intervals: $(-\infty,-2)$, $(-2,7)$, and $(7,\infty)$
We can test a value in each interval to see if it satisfies the inequality
For $x=-3$, we have $(-3)^2-5(-3)-4=9+15-4=20$, which is not less than or equal to 10
For $x=0$, we have $0^2-5(0)-4=-4$, which is less than or equal to 10
For $x=8$, we have $8^2-5(8)-4=64-40-4=20$, which is not less than or equal to 10
So the solution is the interval where the inequality holds, which is $\boxed{(-2,7)}$
ANSWER: \boxed{(-2,7)}

EXPERT COMMENTARY 2 0:

The explanation provided is correct and clear in its approach to solving the inequality. The steps are well laid out and the reasoning is sound.

One possible improvement would be to mention that the quadratic equation $x^2-5x-4=10$ is first rearranged to $x^2-5x-14=0$ before factoring. This would make the process more understandable for students who might not be familiar with directly solving the equation in its given form.

The method of testing a value in each interval to determine the solution is a valid and systematic approach. It's good that the explanation includes this step as it provides a way to confirm the solution.

Overall, the solution is correctly obtained and the reasoning is well-explained. The use of interval notation to express the final answer is also appropriate and clear.

The explanation is clear, concise, and logically sound. It effectively guides the reader through the process of solving the inequality.

EXPERT COMMENTARY 2 1:

The expert's commentary is mostly accurate and provides a positive assessment of the given explanation. However, there are a few points that need to be addressed to improve the clarity and correctness of the explanation.

1. The expert correctly points out that rearranging the given quadratic equation $x^2-5x-4=10$ to $x^2-5x-14=0$ before factoring would make the process more understandable. This is a good suggestion for clarity, especially for students who might be less familiar with solving quadratic equations directly.

2. The method of testing a value in each interval to determine the solution is indeed a valid and systematic approach. However, it would be beneficial to explain why this method works. By mentioning that the inequality $x^2-5x-4 \le 10$ represents a parabola opening upwards, and testing points in each interval helps determine where the parabola is below or at the level of $10$, the explanation would be more complete.

3. It would be helpful to add a concluding statement explicitly stating that the solution is the interval where the inequality holds, which is $(-2,7)$, and then express this in interval notation.

Overall, the given explanation is clear, logically sound, and effectively guides the reader through the process of solving the inequality. With the suggested improvements, the explanation would be more comprehensive and precise.

EXPERT COMMENTARY 2 2:

The expert's commentary provides a positive assessment of the given explanation, acknowledging its correctness and clarity. However, there are some critical points that need to be addressed to improve the precision and completeness of the explanation.

1. The suggestion to rearrange the quadratic equation $x^2-5x-4=10$ to $x^2-5x-14=0$ before factoring is crucial for clarity. This step should be explicitly included in the reasoning to ensure a complete and understandable solution.

2. While the method of testing a value in each interval to determine the solution is valid, it would be beneficial to explain why this method works. Mentioning that the inequality $x^2-5x-4 \le 10$ represents a parabola opening upwards and testing points in each interval helps determine where the parabola is below or at the level of $10$ would provide a more complete explanation.

3. The conclusion should explicitly state that the solution is the interval where the inequality holds, which is $(-2,7)$, and then express this in interval notation. This final step is essential for a complete and precise solution.

In summary, the given explanation is clear and logically sound, but it would benefit from including the suggested improvements to make it more comprehensive and precise.

EXPLANATION 3:

REASONING:
First we solve the quadratic inequality $x^2-5x-4\leq10$
By subtracting 10 from both sides, we get $x^2-5x-14\leq0$
Factor the quadratic to get $(x-7)(x+2)\leq0$
The solutions to the equation $x^2-5x-4=10$ are $x=7$ and $x=-2$
These solutions divide the real number line into 3 intervals: $(-\infty,-2]$, $[-2,7]$, and $[7,\infty)$
We then test a value in each interval, e.g. $x=-3$, $x=0$, and $x=8$
We find that the inequality is true for $x\in\boxed{[-2,7]}$
ANSWER: \boxed{[-2,7]}

EXPERT COMMENTARY 3 0:

The initial steps are correct: subtracting 10 from both sides and factoring the quadratic inequality. However, the conclusion is incorrect because the inequality $(x-7)(x+2) \leq 0$ is not equivalent to the equation $x^2 - 5x - 4 = 10$. 

To find the correct solution, we need to determine the sign of the quadratic expression $(x-7)(x+2)$ in the intervals defined by the roots of the equation $x^2-5x-4=10$. This can be done by considering the signs of the factors $(x-7)$ and $(x+2)$ separately.

1. Consider the interval $(-\infty, -2)$. 
   - Test a value, e.g. $x=-3$, in the inequality $(x-7)(x+2) \leq 0$.
   - Determine that $(x-7)$ is negative and $(x+2)$ is negative, so their product is positive.

2. Consider the interval $(-2, 7)$.
   - Test a value, e.g. $x=0$, in the inequality $(x-7)(x+2) \leq 0$.
   - Determine that $(x-7)$ is negative and $(x+2)$ is positive, so their product is negative.

3. Consider the interval $(7, \infty)$.
   - Test a value, e.g. $x=8$, in the inequality $(x-7)(x+2) \leq 0$.
   - Determine that $(x-7)$ is positive and $(x+2)$ is positive, so their product is positive.

Therefore, the solution to the inequality $x^2-5x-4\leq10$ is $x\in\boxed{[-2,7]}$, as the inequality is true when $x$ is in that interval.

EXPERT COMMENTARY 3 1:

The initial reasoning is incorrect because it incorrectly equates the inequality $(x-7)(x+2) \leq 0$ to the equation $x^2 - 5x - 4 = 10$. 
We can correct the explanation by considering the sign of the quadratic expression $(x-7)(x+2)$ in the intervals defined by the roots of the equation $x^2-5x-4=10$. 
1. Consider the interval $(-\infty, -2)$. Test a value, e.g. $x=-3$, in the inequality $(x-7)(x+2) \leq 0$. Determine that $(x-7)$ is negative and $(x+2)$ is negative, so their product is positive.
2. Consider the interval $(-2, 7)$. Test a value, e.g. $x=0$, in the inequality $(x-7)(x+2) \leq 0$. Determine that $(x-7)$ is negative and $(x+2)$ is positive, so their product is negative.
3. Consider the interval $(7, \infty)$. Test a value, e.g. $x=8$, in the inequality $(x-7)(x+2) \leq 0$. Determine that $(x-7)$ is positive and $(x+2)$ is positive, so their product is positive.

Therefore, the correct solution to the inequality $x^2-5x-4\leq10$ is $x\in\boxed{[-2,7]}$. The inequality is true when $x$ is in that interval.

EXPERT COMMENTARY 3 2:

The initial reasoning incorrectly equates the inequality $(x-7)(x+2) \leq 0$ to the equation $x^2 - 5x - 4 = 10$. 
We can correct the explanation by considering the sign of the quadratic expression $(x-7)(x+2)$ in the intervals defined by the roots of the equation $x^2-5x-4=10$. 
1. Consider the interval $(-\infty, -2)$. Test a value, e.g. $x=-3$, in the inequality $(x-7)(x+2) \leq 0$. Determine that $(x-7)$ is negative and $(x+2)$ is negative, so their product is positive.
2. Consider the interval $(-2, 7)$. Test a value, e.g. $x=0$, in the inequality $(x-7)(x+2) \leq 0$. Determine that $(x-7)$ is negative and $(x+2)$ is positive, so their product is negative.
3. Consider the interval $(7, \infty)$. Test a value, e.g. $x=8$, in the inequality $(x-7)(x+2) \leq 0$. Determine that $(x-7)$ is positive and $(x+2)$ is positive, so their product is positive.

Therefore, the correct solution to the inequality $x^2-5x-4\leq10$ is $x\in\boxed{[-2,7]}$. The inequality is true when $x$ is in that interval.

The explanations given by Expert 0 and Expert 1 are correct and improve the initial reasoning. They provide a clear and precise explanation of how to correctly solve the inequality using the sign of the quadratic expression. The criticisms are valid and help to refine the initial approach.

Propose the best possible explanation and answer.