[ Absolute value ]

Solve the equation $|x-2|+|x-1|+|x|+|x+1|+|x+2|=6$.

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The number |x-a| is equal to the distance from the point on the number line with coordinate x to the point with coordinate a.

## Solution

Consider a point on the number line with coordinate x. The sum $|\mathrm{x}-2|+|\mathrm{x}-1|+|\mathrm{x}|+|\mathrm{x}+1|+|\mathrm{x}+2|$ is equal to the sum of the distances from the point x to the points with coordinates $2,1,0,-1,-2$. Note that the sum of the distances from any point to points A and B is not less than the length of the segment AB (and equality is achieved if and only if the point is located on the segment AB). From this, we get that |x-2|+|x+2| is not less than 4, and |x-1|+|x+1| is not less than 2 for any

| x. Therefore, for the sum $|x-2|+|x-1|+|x|+|x+1|+|x+2|$ to be equal to 2+4=6, it is necessary that |x|=0. Thus, x must be equal to 0. It is easy to verify that the value $x=0$ is indeed a solution to the given equation.

## Answer

$\mathrm{x}=0$.