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Italian Mathematical Olympiad

Algebra Difficulty 3.8 AMC 10/12 Find the answer Italy

Problem:

As the real parameter $a$ varies, what is the maximum number of solutions of the equation $||x-1|-4|+x=a$ ?

Pick one

Solution

Solution:

The answer is (E). Indeed, for every $x \leq -3$ , $x-1 \leq 0$ and hence $|x-1| = 1-x$ , moreover $|x-1|-4 = -x-3 \geq 0$ , and the equation can be simplified as follows:
$||x-1|-4|+x = |-x+1-4|+x = -x-3+x = -3$
Therefore if $a = -3$ the equation has infinitely many solutions.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.