1960 AMC 12 Problem 26

Attempt Problem 26 of the 1960 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1960 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

26.

Find the set of xx-values satisfying the inequality ∣5−x3∣<2. \left|\frac{5-x}{3}\right|\lt2. [The symbol ∣a∣|a| means +a+a if aa is positive, −a-a if aa is negative, 00 if aa is zero. The notation 1<a<21\lt a\lt2 means that aa can have any value between 11 and 2,2, excluding 11 and 2.2.]
1<x<111\lt x\lt11
−1<x<11-1\lt x\lt11
x<11x\lt11
x>11x\gt11
∣x∣<6|x|\lt6
Answer: B
Concepts:absolute valueinequality
Small hint:
Multiply the inequality by 33
Big hint:
Rewrite ∣5−x∣<6|5-x|\lt6 as a compound inequality
Solution:
The inequality is equivalent to ∣5−x∣<6, |5-x|\lt6, so −6<5−x<6.-6\lt5-x\lt6. Subtracting 55 and reversing signs as needed gives −1<x<11. -1\lt x\lt11.
Therefore, the correct answer is B.
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