Solve the Linear-Rational Equation: 11x - 1/x = 10

Let's solve the given equation:

$11x-\frac{1}{x}=10$

Begin by eliminating the fraction line on the left side of the given equation. We can achieve this by multiplying both sides of the equation by the common denominator - which is the unknown $x$. Note that the denominator (before multiplying by the common denominator - meaning in the given equation) cannot be zero since the fraction would be undefined. Therefore we must always define the domain accordingly, and the denominator should not be zero(mentioned in the first line of the solution below) This step is a mandatory step when solving an equation:

$11x-\frac{1}{x}=10 \hspace{8pt}\text{/}\cdot x\hspace{8pt}\Leftrightarrow \boxed{ x\neq0}\\ x\cdot 11x-1\cdot 1 =x\cdot10 \\ 11x^2-1=10x\\ 11x^2-10x-1=0\\$In the final stage after obtaining a quadratic equation where the coefficient of the first-degree term (of the unknown) is not zero (meaning such a term exists in the equation), we moved all terms to one side,

From here, we'll proceed to solve the expression using the quadratic formula.

Let's recall the quadratic formula:

The rule states that for a quadratic equation in the general form:

$ax^2+bx+c =0$

there are two solutions (or fewer) which we find using the formula:

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Let's return now to the equation that we obtained in the last stage:

$11x^2-10x-1=0$

Note the coefficients from the general form that we mentioned in the rule above:

$ax^2+bx+c$are:

$\begin{cases} a=11\\ b=-10\\ c=-1 \\ \end{cases}$

We didn't forget to consider the coefficient together with its sign,

Therefore the solutions to the quadratic equation we obtained in the last stage are:

$x_{1,2}=\frac{-(-10)\pm\sqrt{(-10)^2-4\cdot11\cdot(-1)}}{2\cdot11}\\ x_{1,2}=\frac{10\pm\sqrt{100+44}}{22}\\ x_{1,2}=\frac{10\pm12}{22}\\ \downarrow\\ x_{1,2}=\frac{22}{22},\hspace{4pt}\frac{-2}{22}\\ \boxed{x=1,\hspace{4pt}-\frac{1}{11}}$

In the final stage we simplified the fractions that were obtained as solutions,

Let's summarize then the solution of the equation:

$11x-\frac{1}{x}=10 \hspace{8pt}\text{/}\cdot x\hspace{8pt}\Leftrightarrow \boxed{ x\neq0}\\ 11x^2-1=10x\\ 11x^2-10x-1=0\\ \boxed{x=1,\hspace{4pt}-\frac{1}{11}}$

Note that both solutions we obtained for the unknown in the equation do not contradict the domain that was specified and therefore both are valid.

Therefore the correct answer is answer B.