Multiple choice

For the equation $|x-1|+|x-3|-2|x-2|=\lambda x$, which of the following is/are correct

  1. For $\lambda=1$, then equation has $2$ distinct real solutions.
  2. For $\lambda\in (0, 1)$, the equation has $3$ distinct real roots
  3. For $\lambda\in (1, 2)$, the equation has $3$ distinct real roots
  4. For $\lambda=0$, the equation has infinite solutions.
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A Correct answer
AI explanation

We evaluate the piecewise function y = |x-1| + |x-3| - 2|x-2| across its critical intervals to find it forms a peak at the point (2, 2). For lambda = 1, the equation becomes y = x, which is a line passing through the origin with a slope of 1. This straight line intersects the bounded peak of the function at exactly two distinct points, meaning the equation has two distinct real solutions.

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