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| − | ==Problem==
| + | #REDIRECT [[2021 AMC 12B Problems/Problem 1|Solution]] |
| − | How many integer values of <math>x</math> satisfy <math>|x|<3\pi</math>?
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| − | <math>\textbf{(A)} ~9 \qquad\textbf{(B)} ~10 \qquad\textbf{(C)} ~18 \qquad\textbf{(D)} ~19 \qquad\textbf{(E)} ~20</math>
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| − | ==Solution 1==
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| − | Since <math>3\pi</math> is about <math>9.42</math>, we multiply 9 by 2 and add 1 to get <math> \boxed{\textbf{(D)}\ ~19} </math>~smarty101
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| − | ==Solution 2==
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| − | <math>3\pi \approx 9.4.</math> There are two cases here.
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| − | When <math>x>0, |x|>0,</math> and <math>x = |x|.</math> So then <math>x<9.4</math>
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| − | When <math>x<0, |x|>0,</math> and <math>x = -|x|.</math> So then <math>-x<9.4</math>. Dividing by <math>-1</math> and flipping the sign, we get <math>x>-9.4.</math>
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| − | From case 1 and 2, we need <math>-9.4 < x < 9.4</math>. Since <math>x</math> is an integer, we must have <math>x</math> between <math>-9</math> and <math>9</math>. There are a total of <cmath>9-(-9) + 1 = \boxed{\textbf{(D)}\ ~19} \text{ integers}.</cmath>
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| − | -PureSwag
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| − | ==Solution 3==
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| − | <math>|x|<3\pi</math> <math>\iff</math> <math>-3\pi<x<3\pi</math>. Since <math>\pi</math> is approximately <math>3.14</math>, <math>3\pi</math> is approximately <math>9.42</math>. We are trying to solve for <math>-9.42<x<9.42</math>, where <math>x\in\mathbb{Z}</math>. Hence, <math>-9.42<x<9.42</math> <math>\implies</math> <math>-9\leq x\leq9</math>, for <math>x\in\mathbb{Z}</math>. The number of integer values of <math>x</math> is <math>9-(-9)+1=19</math>. Therefore, the answer is <math>\boxed{\textbf{(D)}19}</math>.
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| − | <br><br>
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| − | ~ {TSun} ~
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| − | {{AMC10 box|year=2021|ab=B|before=First Problem|num-a=2}}
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