1964 AMC 12 Problem 27

Attempt Problem 27 of the 1964 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 1964 AMC 12 solutions, or check the answer key.

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

27.

If $x$ is a real number and $|x-4|+|x-3|\lt a,$ where $a\gt0,$ then:

$0\lt a\lt0.01$

$0.01\lt a\lt1$

$0\lt a\lt1$

$0\lt a\leq1$

$a\gt1$

Answer: E

Concepts:absolute value triangle inequality inequality

Difficulty rating: 1260

Small Hint:

Interpret the sum as the distances from $x$ to $3$ and $4$

Big Hint:

Find its minimum for $3\leq x\leq4$ , remembering the inequality is strict

Solution:

By the triangle inequality, $|x-4|+|x-3|\geq|4-3|=1.$ Equality holds for every $x$ between $3$ and $4.$ Therefore the strict inequality has a real solution exactly when $a\gt1.$

Thus, the correct answer is E.

← Problem 26#26

Full Exam

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