If you define the convex hull of a set $X$ as the set of all convex combinations of elements of $X$, it becomes difficult to decide if a given element $w$ belongs or not to $conv(X)$ (You have to discover whether $w$ can be written as a convex combination of elements of X or not). But if you have a characterization of $conv(X)$ as a system of (in)equalities, it becomes easier.
Consider the sets:
$$A=\{(x,y,z)\in\mathbb{R}^3; y\geq1, z\geq1, y+z=3, x=3\} ,$$ $$B=\{(x,y,z)\in\mathbb{R}^3; x\geq1, y\geq1, x+y=3, z=3\}, $$ $$C=\{(x,y,z)\in\mathbb{R}^3; x\geq1, z\geq1, x+z=3, y=3\}. $$
These three sets are segments. Show that the convex hull $\operatorname{conv}(A\cup B\cup C)$ is equal the set:
$$\begin{cases} x\geq1,y\geq 1,z\geq 1; \\ x+y\geq3, y+z\geq3, x+z\geq3;\\ x+y+z=6 \end{cases} $$