| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | B. -14 |
B |
| deepseekr1 | The problem requires finding the sum of all integer solutions to the inequality \( |n| < |n - 3| < 9 \). 1. **Solve \( |n - 3| < 9 \):** - This simplifies to \( -9 < n - 3 < 9 \), leading to \( -6 < n < 12 \). Integer solutions are \( n = -5, -4, \ldots, 11 \). 2. **Solve \( |n| < |n - 3| \):** - Squaring both sides: \( n^2 < (n - 3)^2 \), which simplifies to \( 0 < -6n + 9 \). Solving gives \( n < 1.5 \). For integers, \( n \leq 1 \). 3. **Intersection of solutions:** - Combining \( -6 < n < 12 \) and \( n \leq 1 \), valid integers are \( n = -5, -4, -3, -2, -1, 0, 1 \). 4. **Sum the solutions:** - Summing \( -5 + (-4) + (-3) + (-2) + (-1) + 0 + 1 \) yields \( -14 \). **Answer:** B. -14 |
B |
| gpto1 | To determine the sum of all integer solutions to the inequality \(|n| < |n - 3| < 9\), we will solve the inequality step by step. **First inequality:** \(|n - 3| < 9\) This inequality can be rewritten without absolute values by considering the definition of absolute value: \[ -9 < n - 3 < 9 \] Adding 3 to all parts: \[ -6 < n < 12 \] So \(n\) can be any integer from \(-5\) to \(11\) inclusive. **Second inequality:** \(|n| < |n - 3|\) We will check this inequality for each integer \(n\) from \(-5\) to \(11\): 1. **\(n = -5\):** - \(|-5| = 5\) - \(|-5 - 3| = |-8| = 8\) - \(5 < 8\) → True 2. **\(n = -4\):** - \(|-4| = 4\) - \(|-4 - 3| = |-7| = 7\) - \(4 < 7\) → True 3. **\(n = -3\):** - \(|-3| = 3\) - \(|-3 - 3| = |-6| = 6\) - \(3 < 6\) → True 4. **\(n = -2\):** - \(|-2| = 2\) - \(|-2 - 3| = |-5| = 5\) - \(2 < 5\) → True 5. **\(n = -1\):** - \(|-1| = 1\) - \(|-1 - 3| = |-4| = 4\) - \(1 < 4\) → True 6. **\(n = 0\):** - \(|0| = 0\) - \(|0 - 3| = |-3| = 3\) - \(0 < 3\) → True 7. **\(n = 1\):** - \(|1| = 1\) - \(|1 - 3| = |-2| = 2\) - \(1 < 2\) → True 8. **\(n = 2\):** - \(|2| = 2\) - \(|2 - 3| = |-1| = 1\) - \(2 < 1\) → False 9. **For \(n \geq 2\):** - The inequality \(|n| < |n - 3|\) is not satisfied. **Conclusion:** The integer values of \(n\) that satisfy both inequalities are \(-5, -4, -3, -2, -1, 0, 1\). **Sum of these integers:** \[ (-5) + (-4) + (-3) + (-2) + (-1) + 0 + 1 = -14 \] **Answer:** **B. -14** |
B |
| GPT-4o | To solve the inequality \( |n| < |n-3| < 9 \), we need to consider the properties of absolute values and solve the inequalities step by step. 1. \( |n| < |n-3| \) To solve this, we need to consider two cases for \( n \): **Case 1: \( n \geq 0 \)** If \( n \geq 0 \), \( |n| = n \), and we need to solve \( n < |n-3| \). There are two sub-cases here: - \( n < n - 3 \): This is impossible as it simplifies to \( 0 < -3 \), which is false. - \( n < 3 - n \): Solving this, we get \( 2n < 3 \) or \( n < \frac{3}{2} \). Since \( n \geq 0 \) and \( n < \frac{3}{2} \), \( n \) can be \( 0, 1 \). **Case 2: \( n < 0 \)** If \( n < 0 \), \( |n| = -n \), and we need to solve \( -n < |n-3| \). There are two sub-cases here: - \( -n < n - 3 \): Solving this, we get \( 0 < 2n - 3 \) or \( 3 < 2n \), which simplifies to \( n > \frac{3}{2} \). This is impossible as \( n \) is negative. - \( -n < 3 - n \): Solving this, we get \( 0 < 3 \), which is always true. Thus, for \( n < 0 \), \( n \) can be any negative integer. Combining both cases, the integer solutions from this part are \( n \in \{0, 1\} \cup \{n < 0\} \). 2. \( |n-3| < 9 \) This inequality means that \( -9 < n-3 < 9 \). Solving for \( n \), we get: \( -6 < n < 12 \) So the integer solutions from this part are \( n \in \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \). Combining both parts, we need the integers that satisfy both inequalities: \( n \in \{-5, -4, -3, -2, -1, 0, 1\} \) Finally, summing these solutions: \[ -5 + (-4) + (-3) + (-2) + (-1) + 0 + 1 = -14 \] Therefore, the sum of all integer solutions is \(-14\). Answer: B. |
B |
| llama3.1-70B-Instruct | B |
B |
| llama3.1-405B-Instruct | We have $|n-3|<9$, so $-9 < n-3 < 9$, which implies $-6 < n < 12$. We also have $|n| < |n-3|$, so $n^2 < (n-3)^2$, which implies $0 < -6n + 9$, so $n < \frac{3}{2}$. Therefore, the solutions to $|n| < |n-3| < 9$ are $-5$, $-4$, $-3$, $-2$, $-1$, $0$, and $1$, and the sum of these solutions is $\boxed{\textbf{(B) }-14}.$ The best answer is B. |
B |
| DeepSeek-Coder-V2 | B. -14 | B |
| Qwen2-72B-Instruct | B. -14 | B |
| baichuan4 | B | B |
| doubao | Solve the inequality $|n| \lt |n-3|$. Squaring both sides gives $n^2 \lt (n-3)^2$, $n^2 \lt n^2-6n+9$, $6n \lt 9$, $n \lt \frac{3}{2}$. Solve the inequality $|n-3| \lt 9$. $-9 \lt n-3 \lt 9$, $-6 \lt n \lt 12$. So the integer solutions of $n$ are $-5,-4,-3,-2,-1,0,1$. The sum of these integers is $-14$. So the answer is B. |
B |
| minimax | To solve the inequality $|n| < |n-3| < 9$, we need to consider two cases for the absolute value expressions: when $n$ is positive and when $n$ is negative. Case 1: $n$ is positive or zero. In this case, $|n| = n$ and $|n-3| = n-3$ (since both $n$ and $n-3$ are non-negative). The inequality becomes: $$n < n-3 < 9$$ This simplifies to: $$0 < -3 < 9$$ Which is always true for positive $n$. However, we also have the upper bound $n-3 < 9$, which gives us: $$n < 12$$ Combining this with the lower bound $n > 0$, we get the range of solutions for positive $n$: $$0 < n < 12$$ The integer solutions in this range are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. Case 2: $n$ is negative. In this case, $|n| = -n$ and $|n-3| = -(n-3) = 3-n$ (since $n$ is negative and $n-3$ is more negative). The inequality becomes: $$-n < 3-n < 9$$ This simplifies to: $$0 > -3 > -9$$ Which is always true for negative $n$. However, we also have the upper bound $3-n < 9$, which gives us: $$n > -6$$ Combining this with the lower bound $n < 0$, we get the range of solutions for negative $n$: $$-6 < n < 0$$ The integer solutions in this range are: -5, -4, -3, -2, -1. Now, we sum all the integer solutions from both cases: $$(-5) + (-4) + (-3) + (-2) + (-1) + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11$$ $$= -15 + 55$$ $$= 40$$ However, none of the options given (A, B, C, D) match the sum we calculated. It seems there might be an error in the question or the options provided. If we were to choose the closest option, it would be none of them, but if we must choose from the given options, we would need to recheck our calculations or the question itself. |
B |
| kimi | To solve this problem, we need to find the integer solutions to the inequality \( |n| < |n-3| < 9 \). First, let's break down the inequality into two parts: 1. \( |n| < |n-3| \) 2. \( |n-3| < 9 \) For the first part, \( |n| < |n-3| \), we can consider two cases: - Case 1: \( n \geq 0 \) and \( n < (n-3) \), which is impossible since \( n \) cannot be less than \( n-3 \). - Case 2: \( n < 0 \) and \( -n < -(n-3) \), which simplifies to \( n < 3-n \) or \( 2n < 3 \) or \( n < \frac{3}{2} \). For the second part, \( |n-3| < 9 \), we have two cases: - Case 1: \( n-3 \geq 0 \) and \( n-3 < 9 \), which simplifies to \( n \geq 3 \) and \( n < 12 \). - Case 2: \( n-3 < 0 \) and \( -(n-3) < 9 \), which simplifies to \( n < 3 \) and \( n > -6 \). Combining the valid ranges from both parts, we get: - From \( |n| < |n-3| \), we have \( n < \frac{3}{2} \). - From \( |n-3| < 9 \), we have \( -6 < n < 12 \). The intersection of these two ranges is \( -6 < n < \frac{3}{2} \). Since we are looking for integer solutions, the possible values for \( n \) are \( -5, -4, -3, -2, -1, 0, 1, 2 \). Now, let's find the sum of these integer solutions: \( (-5) + (-4) + (-3) + (-2) + (-1) + 0 + 1 + 2 = -14 \) So the answer is B. -14. |
B |
| wenxinyiyan | B. -14 Explanation: Given the inequality $|n| < |n-3| < 9$, we can break it down into two parts: 1. $|n| < |n-3|$ 2. $|n-3| < 9$ Solving the first inequality $|n| < |n-3|$: This can be rewritten as $n^2 < (n-3)^2$ (since $|x| = \sqrt{x^2}$ and squaring both sides preserves the inequality). Expanding and simplifying, we get: $n^2 < n^2 - 6n + 9$ $0 < -6n + 9$ $6n < 9$ $n < \frac{3}{2}$ Solving the second inequality $|n-3| < 9$: This can be rewritten as $-9 < n-3 < 9$. Adding 3 to all parts of the inequality, we get: $-6 < n < 12$ Combining the two inequalities, we get: $-6 < n < \frac{3}{2}$ Since $n$ must be an integer, the possible values of $n$ are $-5, -4, -3, -2, -1, 0, 1$. The sum of these integers is $-5 + (-4) + (-3) + (-2) + (-1) + 0 + 1 = -14$. |
B |
| zhipu | To solve the inequality \( |n| < |n-3| < 9 \), we need to analyze it step by step. First, consider the inner inequality \( |n| < |n-3| \). ### Step 1: Analyze \( |n| < |n-3| \) This inequality implies that the distance of \( n \) from 0 is less than the distance of \( n \) from 3. We can break this into two cases based on the definition of absolute value: **Case 1: \( n \geq 0 \)** Here, \( |n| = n \) and \( |n-3| = 3 - n \) if \( n < 3 \), or \( |n-3| = n - 3 \) if \( n \geq 3 \). - If \( 0 \leq n < 3 \): \[ n < 3 - n \implies 2n < 3 \implies n < 1.5 \] Since \( n \) is an integer, \( n \) can be 0 or 1. - If \( n \geq 3 \): \[ n < n - 3 \text{ is never true} \] So, there are no solutions in this range. **Case 2: \( n < 0 \)** Here, \( |n| = -n \) and \( |n-3| = 3 - n \). - If \( n < 0 \): \[ -n < 3 - n \implies 0 < 3 \text{ (always true)} \] So, all negative integers satisfy \( |n| < |n-3| \). ### Step 2: Combine with \( |n-3| < 9 \) Now, we need to consider the outer inequality \( |n-3| < 9 \). This can be written as: \[ -9 < n - 3 < 9 \implies -6 < n < 12 \] ### Step 3: Combine Both Inequalities From the above analysis: - For \( n \geq 0 \), valid integers are \( n = 0, 1 \). - For \( n < 0 \), all integers within the range \( -6 < n < 0 \) are valid, which are \( n = -1, -2, -3, -4, -5 \). Combining these with \( -6 < n < 12 \): - Valid integers are \( -5, -4, -3, -2, -1, 0, 1 \). ### Step 4: Sum of All Valid Integers Summing these integers: \[ -5 + -4 + -3 + -2 + -1 + 0 + 1 = -14 \] Thus, the sum of all integer solutions to \( |n| < |n-3| < 9 \) is \(-14\). **Answer: B. -14** |
B |
| gpto1 | To determine the sum of all integer solutions to the inequality \(|n| < |n - 3| < 9\), we will solve the inequality step by step. **First inequality:** \(|n - 3| < 9\) This inequality can be rewritten without absolute values by considering the definition of absolute value: \[ -9 < n - 3 < 9 \] Adding 3 to all parts: \[ -6 < n < 12 \] So \(n\) can be any integer from \(-5\) to \(11\) inclusive. **Second inequality:** \(|n| < |n - 3|\) We will check this inequality for each integer \(n\) from \(-5\) to \(11\): 1. **\(n = -5\):** - \(|-5| = 5\) - \(|-5 - 3| = |-8| = 8\) - \(5 < 8\) → True 2. **\(n = -4\):** - \(|-4| = 4\) - \(|-4 - 3| = |-7| = 7\) - \(4 < 7\) → True 3. **\(n = -3\):** - \(|-3| = 3\) - \(|-3 - 3| = |-6| = 6\) - \(3 < 6\) → True 4. **\(n = -2\):** - \(|-2| = 2\) - \(|-2 - 3| = |-5| = 5\) - \(2 < 5\) → True 5. **\(n = -1\):** - \(|-1| = 1\) - \(|-1 - 3| = |-4| = 4\) - \(1 < 4\) → True 6. **\(n = 0\):** - \(|0| = 0\) - \(|0 - 3| = |-3| = 3\) - \(0 < 3\) → True 7. **\(n = 1\):** - \(|1| = 1\) - \(|1 - 3| = |-2| = 2\) - \(1 < 2\) → True 8. **\(n = 2\):** - \(|2| = 2\) - \(|2 - 3| = |-1| = 1\) - \(2 < 1\) → False 9. **For \(n \geq 2\):** - The inequality \(|n| < |n - 3|\) is not satisfied. **Conclusion:** The integer values of \(n\) that satisfy both inequalities are \(-5, -4, -3, -2, -1, 0, 1\). **Sum of these integers:** \[ (-5) + (-4) + (-3) + (-2) + (-1) + 0 + 1 = -14 \] **Answer:** **B. -14** |
B |
| deepseekv3 | We are tasked with finding all integer solutions to the inequality \( |n| < |n - 3| < 9 \) and then calculating their sum. --- ### Step 1: Solve \( |n| < |n - 3| \) The inequality \( |n| < |n - 3| \) implies that \( n \) is closer to 0 than to 3 on the number line. Geometrically, this occurs when \( n < 1.5 \). Since \( n \) is an integer, the solutions are: \[ n \leq 1 \] --- ### Step 2: Solve \( |n - 3| < 9 \) The inequality \( |n - 3| < 9 \) implies \( -9 < n - 3 < 9 \), or: \[ -6 < n < 12 \] --- ### Step 3: Combine the inequalities From Step 1 and Step 2, the combined inequality is: \[ -6 < n \leq 1 \] The integer solutions are: \[ n = -5, -4, -3, -2, -1, 0, 1 \] --- ### Step 4: Calculate the sum The sum of the integer solutions is: \[ -5 + (-4) + (-3) + (-2) + (-1) + 0 + 1 = -14 \] --- The correct answer is **B. -14**. |
B |