2021 AMC 10B Problems/Problem 1

Problem

How many integer values of $x$ satisfy $|x|<3\pi$ ?

$\textbf{(A)} ~9 \qquad\textbf{(B)} ~10 \qquad\textbf{(C)} ~18 \qquad\textbf{(D)} ~19 \qquad\textbf{(E)} ~20$

Solution

Since $3\pi$ is about $9.42$ , we multiply 9 by 2 and add 1 to get $\boxed{\textbf{(D)}\ ~19}$ ~smarty101

Solution 2

$3\pi \approx 9.4.$ There are two cases here.

When $x>0, |x|>0,$ and $x = |x|.$ So then $x<9.4$

When $x<0, |x|>0,$ and $x = -|x|.$ So then $-x<9.4$ . Dividing by $-1$ and flipping the sign, we get $x>-9.4.$

From case 1 and 2, we need $-9.4 < x < 9.4$ . Since $x$ is an integer, we must have $x$ between $-9$ and $9$ . There are a total of $9-(-9) + 1 = \boxed{\textbf{(D)}\ ~19} \text{ integers}.$

-PureSwag

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2021 AMC 10B Problems/Problem 1

Problem

Solution

Solution 2