College Algebra, Chapter 1, 1.1, Section 1.1, Problem 8

Check whether a.) $x = 2$ or b.) $x = 4$ is a solution of the equation $\displaystyle \frac{1}{x} - \frac{1}{x-4} = 1$

a.) $x = 2$

$\begin{equation} \begin{aligned} \frac{1}{(2)} - \frac{1}{(2) - 4} &= 1 && \text{Substitute } x = 2\\ \\ \frac{1}{2} - \frac{1}{-2} &= 1 && \text{Simplify}\\ \\ \frac{1+1}{2} &= 1 && \text{Get the LCD}\\ \\ \frac{2}{2} &= 1 && \text{Simplifty} \\ \\ 1 &= 1 \end{aligned} \end{equation}$

So $x = 2$ is the solution to the equation.

b.) $x = 4$

$\begin{equation} \begin{aligned} \frac{1}{(4)} - \frac{1}{(4) -4} &=1 && \text{Substitute } x =4\\ \\ \frac{1}{4} - \frac{1}{4-4} &= 1 && \text{Simplify}\\ \\ \frac{1}{4} - \frac{1}{0} &=1 && \text{Equation is undefined, denomitor is zero} \end{aligned} \end{equation}$

So $x = 4$ is not the solution to the equation.