College Algebra, Chapter 1, 1.1, Section 1.1, Problem 50

The equation $\displaystyle \frac{1}{x} - \frac{2}{2x+1} = \frac{1}{2x^2 + x}$ is either linear or equivalent to a linear equation. Solve the equation

$\begin{equation} \begin{aligned} \frac{1}{x} - \frac{2}{2x+1} &= \frac{1}{2x^2 + x} && \text{Get the LCD}\\ \\ \frac{\cancel{2}x+1 - \cancel{2}x}{x(2x+1)} &= \frac{1}{2x^2 + x} && \text{Combine like terms}\\ \\ \frac{1}{2x^2+x} &= \frac{1}{2x^2+x} \end{aligned} \end{equation}$


It shows that the equation on the left side is equal to the equation on the right side and that every value of $x$ is a solution. So the two equations will have the same graph.