# How do you solve x/3 - 2/3 = 1/x?

May 27, 2016

$x = - 1$ or $x = 3$

#### Explanation:

First we want to eliminate all of the denominators. To do so, multiply everything by $3 x$ (product of all denominators).

$3 x \left(\frac{x}{3}\right) - 3 x \left(\frac{2}{3}\right) = 3 x \left(\frac{1}{x}\right)$

${x}^{2} - 2 x = 3$

Move the $3$ to the other side so that everything equals zero

${x}^{2} - 2 x - 3 = 0$

This is factorable. We need numbers that add to $- 2$ and multiply to $- 3$

$\left(x + 1\right) \left(x - 3\right) = 0$

Ergo, $x$ is either $- 1$ or $3$. Test by subbing into the original equation

$- \frac{1}{3} - \frac{2}{3} = \frac{1}{-} 1$

$- \frac{3}{3} = \frac{1}{-} 1$

yes!

$\frac{3}{3} - \frac{2}{3} = \frac{1}{3}$

yes!