2. For which real numbers $a$ does the equation

$$
|x-2|+|3-x|=a
$$

have exactly two solutions?
Solution.

First, let's sketch the curve $y=|x-2|+|3-x|$.

For $x<2$, $y=2-x+3-x=5-2x$.

For $2 \leq x \leq 3$, $y=x-2+3-x=1$.

For $x>3$, $y=x-2-3+x=2x-5$.

![](https://cdn.mathpix.com/cropped/2024_05_30_d793ff9f9da41a4dd093g-06.jpg?height=780&width=1031&top_left_y=227&top_left_x=661)

The equation $|x-2|+|3-x|=a$ has exactly two solutions for those values of $a$ for which the line $y=a$ intersects the curve at exactly two points, i.e., for $a>1$.