Find the minimum value of $|x-2|+|2x+1|-|x-1|$

Question

Find the minimum value of $$|x-2|+|2x+1|-|x-1|$$

My way of doing it

$|x|$ is also known as distance of x from the origin

$|x-2|$ distance from the origin will be $2$

$|2x+1|$ distance from the origin will be $-1/2$

$|x-1|$ distance from the origin will be $1$

Now in order to find the minimum value, I assumed that on the number line.

Suppose $A=2$, $B=-1/2$ and $C=1$ are friends. Now they all want to meet at a point on the number line where minimum distance is covered. That minimum point will be 1.

Now in order to find the minimum value, I know A will cover $1 unit$ to reach C and B will cover $3/2 \text{ units }$ to reach C.

Total distance covered is $5/2$. But, the correct answer is 1. What is wrong in my way of doing it?

Answer 1

The value of $x \in \mathbb{R}$ such that $f(x)$ is minimal belong the set $A=\{2,-1/2,1\}$ (these values are the values such that vanish each module individually). But $f(2)=f(1)=4$ and $f(-1/2)=1$, thus the minimal value of function is $1$.

Answer 2

Your working is about minimizing

$$|x-2|+\left|x+\frac12\right|+|x-1|.$$

In that case, each term is indeed a distance and you are minimizing a sum of distance. Modeling of imagining $3$ friends picking up a common meeting place to minimize total travel distance holds.

However, if your question is

$$|x-2|+\left|2x+1\right|+|x-1|,$$

then you have to imagine that you have $4$ friends as the question now can be viewed as

$$|x-2|+2\left|x+\frac12\right|+|x-1|=|x-2|+\left|x+\frac12\right|+\left|x+\frac12\right|+|x-1|$$

If your question is indeed

$$|x-2|+\left|2x+1\right|-|x-1|,$$

then we can adopt Alex Provost's suggestion by examining those $3$ points. In that case, we can't visualize it as a sum of distance, in fact, the more the person at $1$ travel, the smaller is the overall value and the modeling of minimizing of sum of travel distance falls apart.