## Task A-1.1.

Determine the minimum value of the expression

$$
|x+1|+|x-2|+|x-3|,
$$

where $x$ is a real number.
## Solution.

Let's calculate the value of the given expression depending on where the number $x$ is located on the real number line.

- For $x \leqslant -1$, the given expression is equal to

$$
-x-1-x+2-x+3=4-3x
$$

We conclude that for real numbers $x \leqslant -1$, the given expression achieves its minimum value when $x$ is the largest, i.e., at the point $x=-1$ and this value is equal to 7.

- For $-1 \leqslant x \leqslant 2$, the given expression is equal to

$$
x+1-x+2-x+3=6-x
$$

the minimum value is achieved at $x=2$ and is equal to 4.

- For $2 \leqslant x \leqslant 3$, the given expression is equal to

$$
x+1+x-2-x+3=x+2
$$

the minimum value is achieved at $x=2$ and is equal to 4.

- For $3 \leqslant x$, the given expression is equal to

$$
x+1+x-2+x-3=3x-4
$$

the minimum value is achieved at $x=3$ and is equal to 5.

Finally, we conclude that the minimum value of the expression is achieved at $x=2$ and is equal to 4.