1. Find the minimum value of the expression $2 x+y$, defined on the set of all pairs $(x, y)$ satisfying the condition

$$
3|x-y|+|2 x-5|=x+1
$$
1.I.1. The following figure shows a set defined by the equation $3|x-y|+|2 x-5|=x+1$.

![](https://cdn.mathpix.com/cropped/2024_05_21_46613852391e49da1fa9g-028.jpg?height=503&width=537&top_left_y=685&top_left_x=751)

Let $C=2 x+y$. The problem requires finding the minimum value of $C$ for all points in the depicted set. Rewriting this relation as $y=C-2 x$, from a geometric perspective, we need to find the smallest $C$ such that the line $y=C-2 x$ intersects the given set. Clearly, the desired value of $C$ is the one for which this line passes through the point $\left(\frac{4}{3} ; \frac{4}{3}\right)$. Thus, $C=\frac{8}{3}+\frac{4}{3}=4$.

Answer: 4.