11.2. Solve the equation $2 \cos (\pi x)=x+\frac{1}{x}$.
Answer: $x=-1$.

Solution. The left side of the equation is no more than two in absolute value, while the right side is no less than two in absolute value: for the right side, this follows from elementary inequalities, for example (for $x>0$), - from the inequality between the arithmetic mean and the geometric mean, and for $x<0$ it is true for the absolute value due to the oddness of the function $y=x+1 / x$; the same conclusion can also be reached by studying the range of values of this function using the derivative. The value of 2 (in absolute value) is only achieved at $x=1$ or $x=-1$. Checking these numbers, we find that only $x=-1$ works.