The equation $x log x = 2 - x$ is satisfied by at least one value of $x$ lying between $1$ and $2$.
The function $f(x) = x log x$ is an increasing function in $[1,2]$ and $g (x)=2-x$ is a decreasing function in $[1, 2]$ and the graphs represented by these functions intersect at a point in $[1,2]$