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

Let's solve the given equation:

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

Let's start by getting rid of the fraction line in the left side of the given equation, we'll do this by multiplying both sides of the equation by the common denominator - which is the unknown $x$, while remembering that in the denominator (before multiplying by the common denominator - meaning in the given equation) cannot be zero since otherwise the fraction is undefined, therefore we must always define the domain accordingly, and require that the denominator is not 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 we identified that we got 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 solve using the quadratic formula, but before we do that in this problem-

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 we got in the last stage:

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

Let's note that the coefficients from the general form we mentioned in the rule above:

$ax^2+bx+c$are:

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

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

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

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

where 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 got 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.