For how many integer values of x is |2x| ≤ 7π?

2024 AMC 12B Problem 3 — Solution

2024 AMC 12B Problem 3

Below is the professionally curated solution for Problem 3 of the 2024 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 12B solutions, or check the answer key.

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Concepts:absolute value counting integers in a range estimation

Difficulty rating: 1130

For how many integer values of $x$ is $|2x| \le 7\pi?$

$16$

$17$

$19$

$20$

$21$

Solution:

The inequality $|2x| \le 7\pi$ is equivalent to $|x| \le \dfrac{7\pi}{2} \approx 10.99.$ The integers satisfying this run from $-10$ to $10,$ which is $10 + 10 + 1 = 21$ values.

Thus, the correct answer is E.

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