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Given $p$: $|4x-3|\leqslant 1$ and $q$: $x^{2}-(2a+1)x+a^{2}+a\leqslant 0$, find the range of values for $a$ if $p$ is a necessary but not sufficient condition for $q$. | [
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"content": "Given $p$: $|4x-3|\\leqslant 1$ and $q$: $x^{2}-(2a+1)x+a^{2}+a\\leqslant 0$, find the range of values for $a$ if $p$ is a necessary but not sufficient condition for $q$.",
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"content": "给定 $p$: $|4x-3|\\leqslant 1$ 和 $q$: $x^{2}-(2a+1)x+a^{2}+a\\leqslant 0$，如果 $p$ 是 ... | 给定 $p$: $|4x-3|\leqslant 1$ 和 $q$: $x^{2}-(2a+1)x+a^{2}+a\leqslant 0$，如果 $p$ 是 $q$ 必要但不充分的条件，则 $a$ 的值范围为 | [0, \\frac{1}{2}] | [] | null | [
"Step 1: Solve inequality \\( p \\): \\( |4x-3| \\leq 1 \\).\nThis can be rewritten as:\n\\[ -1 \\leq 4x - 3 \\leq 1 \\]\nAdding 3 to all parts:\n\\[ 2 \\leq 4x \\leq 4 \\]\nDividing by 4:\n\\[ \\frac{1}{2} \\leq x \\leq 1 \\]"
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"content": "You are a math tutor that helps students solve math problems by breaking them down into clear, logical steps. Follow these guidelines:\n\n# For each step:\n- Clearly explain the reasoning\n- Show the calculated result for any arithmetic... | Given $p$: $|4x-3|\leqslant 1$ and $q$: $x^{2}-(2a+1)x+a^{2}+a\leqslant 0$, find the range of values for $a$ if $p$ is a necessary but not sufficient condition for $q$. Step 1: Solve inequality \( p \): \( |4x-3| \leq 1 \).
This can be rewritten as:
\[ -1 \leq 4x - 3 \leq 1 \]
Adding 3 to all parts:
\[ 2 \leq 4x \leq 4... | Given $p$: $|4x-3|\leqslant 1$ and $q$: $x^{2}-(2a+1)x+a^{2}+a\leqslant 0$, find the range of values for $a$ if $p$ is a necessary but not sufficient condition for $q$. Step 1: Solve inequality \( p \): \( |4x-3| \leq 1 \).
This can be rewritten as:
\[ -1 \leq 4x - 3 \leq 1 \]
Adding 3 to all parts:
\[ 2 \leq 4x \leq 4... |
Given $p$: $|4x-3|\leqslant 1$ and $q$: $x^{2}-(2a+1)x+a^{2}+a\leqslant 0$, find the range of values for $a$ if $p$ is a necessary but not sufficient condition for $q$. | [
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"content": "Given $p$: $|4x-3|\\leqslant 1$ and $q$: $x^{2}-(2a+1)x+a^{2}+a\\leqslant 0$, find the range of values for $a$ if $p$ is a necessary but not sufficient condition for $q$.",
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"content": "给定 $p$: $|4x-3|\\leqslant 1$ 和 $q$: $x^{2}-(2a+1)x+a^{2}+a\\leqslant 0$，如果 $p$ 是 ... | 给定 $p$: $|4x-3|\leqslant 1$ 和 $q$: $x^{2}-(2a+1)x+a^{2}+a\leqslant 0$，如果 $p$ 是 $q$ 必要但不充分的条件，则 $a$ 的值范围为 | [0, \\frac{1}{2}] | [] | null | [
"Step 2: Solve inequality \\( q \\): \\( x^2 - (2a+1)x + a^2 +"
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"content": "You are a math tutor that helps students solve math problems by breaking them down into clear, logical steps. Follow these guidelines:\n\n# For each step:\n- Clearly explain the reasoning\n- Show the calculated result for any arithmetic... | Given $p$: $|4x-3|\leqslant 1$ and $q$: $x^{2}-(2a+1)x+a^{2}+a\leqslant 0$, find the range of values for $a$ if $p$ is a necessary but not sufficient condition for $q$. Step 2: Solve inequality \( q \): \( x^2 - (2a+1)x + a^2 ки | Given $p$: $|4x-3|\leqslant 1$ and $q$: $x^{2}-(2a+1)x+a^{2}+a\leqslant 0$, find the range of values for $a$ if $p$ is a necessary but not sufficient condition for $q$. Step 2: Solve inequality \( q \): \( x^2 - (2a+1)x + a^2 + |
In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $a=4$, $b=5$, and the area of $\triangle ABC$ is $5 \sqrt {3}$, find $|AB|$. | [
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"content": "In acute triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $a=4$, $b=5$, and the area of $\\triangle ABC$ is $5 \\sqrt {3}$, find $|AB|$.",
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"content": "在斜边三角形 $\\triangle ABC$ 中，角 $A$、角 $B$ 和角 $C$ 对应的边分... | 在斜边三角形 $\triangle ABC$ 中，角 $A$、角 $B$ 和角 $C$ 对应的边分别为 $a$, $b$ 和 $c$，如果 $a=4$, $b=5$ 且 $\triangle ABC$ 的面积为 $5\sqrt{3}$，则求 $\mid AB \mid$。 | \\sqrt{21} | [] | null | [
"Step 1: Identify given information: $a=4$, $b=5$, and the area of $\\triangle ABC$ is $5\\sqrt{3}$. We need to find side $c$, which is denoted as $|AB|$.",
"Step 2: The formula for the area of a triangle using two sides and their included angle is $\\frac{1}{2}ab\\sin(C)$ where $C$ is the included angle. Setting... | {
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"content": "You are a math tutor that helps students solve math problems by breaking them down into clear, logical steps. Follow these guidelines:\n\n# For each step:\n- Clearly explain the reasoning\n- Show the calculated result for any arithmetic... | In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $a=4$, $b=5$, and the area of $\triangle ABC$ is $5 \sqrt {3}$, find $|AB|$. Step 1: Identify given information: $a=4$, $b=5$, and the area of $\triangle ABC$ is $5\sqrt{3}$. We need to find side $c... | In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $a=4$, $b=5$, and the area of $\triangle ABC$ is $5 \sqrt {3}$, find $|AB|$. Step 1: Identify given information: $a=4$, $b=5$, and the area of $\triangle ABC$ is $5\sqrt{3}$. We need to find side $c... |
If the linear function is $y=2x-3$, and it is shifted $3$ units upwards, the new equation of the graph is ______. | [
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"content": "If the linear function is $y=2x-3$, and it is shifted $3$ units upwards, the new equation of the graph is ______.",
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"content": "如果线性函数是 $y=2x-3$，并向上移动 $3$ 个单位，那么新的图象方程式为 _______",
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] | 如果线性函数是 $y=2x-3$，并向上移动 $3$ 个单位，那么新的图象方程式为 _______ | y = 2x | [
"Step 1: Understand the effect of shifting a function upwards.\n- Shifting a function upwards by a certain number of units means adding that number to the output (y-value) of the function.",
"Step 2: Apply the shift to the given function.\n- The original function is \\( y = 2x - 3 \\).\n- We need to shift this fu... | null | [
"Step 1: The original equation is y = 2x - 3.",
"Step 2: Shifting the graph 3 units upwards means adding 3 to the y-coordinate of every point on the graph. Therefore, we add 3 to the entire right side of the equation.",
"Step 3: The new equation becomes y = (2x - 3) + 3. Simplifying, we get y = 2x.",
"The ans... | {
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"content": "You are a math tutor that helps students solve math problems by breaking them down into clear, logical steps. Follow these guidelines:\n\n# For each step:\n- Clearly explain the reasoning\n- Show the calculated result for any arithmetic... | If the linear function is $y=2x-3$, and it is shifted $3$ units upwards, the new equation of the graph is ______. Step 1: The original equation is y = 2x - 3ки
Step 2: Shifting the graph 3 units upwards means adding 3 to the y-coordinate of every point on the graph. Therefore, we add 3 to the entire right side of the e... | If the linear function is $y=2x-3$, and it is shifted $3$ units upwards, the new equation of the graph is ______. Step 1: The original equation is y = 2x - 3.
Step 2: Shifting the graph 3 units upwards means adding 3 to the y-coordinate of every point on the graph. Therefore, we add 3 to the entire right side of the eq... |
If the linear function is $y=2x-3$, and it is shifted $3$ units upwards, the new equation of the graph is ______. | [
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"content": "If the linear function is $y=2x-3$, and it is shifted $3$ units upwards, the new equation of the graph is ______.",
"role": "user"
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"content": "如果线性函数是 $y=2x-3$，并向上移动 $3$ 个单位，那么新的图象方程式为 _______",
"role": "assistant"
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] | 如果线性函数是 $y=2x-3$，并向上移动 $3$ 个单位，那么新的图象方程式为 _______ | y = 2x | [
"Step 1: Understand the effect of shifting a function upwards.\n- Shifting a function upwards by a certain number of units means adding that number to the output (y-value) of the function.",
"Step 2: Apply the shift to the given function.\n- The original function is \\( y = 2x - 3 \\).\n- We need to shift this fu... | null | [
"Step 1: Start with the initial linear equation y = 2x - 3.",
"Step 2: To shift"
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"content": "You are a math tutor that helps students solve math problems by breaking them down into clear, logical steps. Follow these guidelines:\n\n# For each step:\n- Clearly explain the reasoning\n- Show the calculated result for any arithmetic... | If the linear function is $y=2x-3$, and it is shifted $3$ units upwards, the new equation of the graph is ______. Step 1: Start with the initial linear equation y = 2x - 3ки
Step 2: To shifки | If the linear function is $y=2x-3$, and it is shifted $3$ units upwards, the new equation of the graph is ______. Step 1: Start with the initial linear equation y = 2x - 3.
Step 2: To shift |
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