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Algebra Difficulty 3.8 AMC 10/12 Find the answer Italy

Problem:

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

Pick one

Solution

Solution:

The answer is (E). Indeed, for every x≤−3x \leq -3, x−1≤0x-1 \leq 0 and hence ∣x−1∣=1−x|x-1| = 1-x, moreover ∣x−1∣−4=−x−3≥0|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 ||x-1|-4|+x = |-x+1-4|+x = -x-3+x = -3
Therefore if a=−3a = -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.