Difference between revisions of "2021 AMC 10B Problems/Problem 1"

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#REDIRECT [[2021 AMC 12B Problems/Problem 1|Solution]]
How many integer values of <math>x</math> satisfy <math>|x|<3\pi</math>?
<math>\textbf{(A)} ~9 \qquad\textbf{(B)} ~10 \qquad\textbf{(C)} ~18 \qquad\textbf{(D)} ~19 \qquad\textbf{(E)} ~20</math>
==Solution 1==
Since <math>3\pi</math> is about <math>9.42</math>, we multiply 9 by 2 for the numbers from <math>1</math> to <math>9</math> and the numbers from <math>-1</math> to <math>-9</math> and add 1 to account for the zero to get <math> \boxed{\textbf{(D)}\ ~19} </math>~smarty101 and edited by Tony_Li2007
==Solution 2==
<math>3\pi \approx 9.4.</math> There are two cases here.
When <math>x>0, |x|>0,</math> and <math>x = |x|.</math> So then <math>x<9.4</math>
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>
From case 1 and 2, we know that <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>
==Solution 3==
<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>.
~ {TSun} ~
==Solution 4==
Looking at the problem, we see that instead of directly saying <math>x</math>, we see that it is <math>|x|.</math> That means all the possible values of <math>x</math> in this case are positive and negative. Rounding <math>\pi</math> to <math>3</math> we get <math>3(3)=9.</math> There are <math>9</math> positive solutions and <math>9</math> negative solutions. <math>9+9=18.</math> But what about zero? Even though zero is neither negative nor positive, but we still need to add it into the solution. Hence, the answer is <math>9+9+1=18+1=\boxed{\textbf{(D)}19}</math>.
==Video Solution 1==
==Video Solution by TheBeautyofMath==
==Video Solution by Interstigation==
==See Also==
{{AMC10 box|year=2021|ab=B|before=First Problem|num-a=2}}
{{AMC12 box|year=2021|ab=B|before=First Problem|num-a=2}}
{{MAA Notice}}

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