How do you solve 2/(x-1 )- 2/3 =4/(x+1)?

Jul 22, 2016

$x = 2 \text{ }$ or $\text{ } x = - 5$

Explanation:

The first thing to note here is that the solution set cannot include $x = \pm 1$, since those values would make the denominators equal to zero.

Now, rearrange the equation to get the all the terms on one side

$\frac{2}{x - 1} - \frac{4}{x + 1} - \frac{2}{3} = 0$

The common denominator here will be

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

which means that you must multiply the first fraction by $1 = \frac{3 \left(x + 1\right)}{3 \left(x + 1\right)}$, the second fraction by $1 = \frac{3 \left(x - 1\right)}{3 \left(x - 1\right)}$, and the third fraction by $1 = \frac{\left(x - 1\right) \left(x + 1\right)}{\left(x - 1\right) \left(x + 1\right)}$ to get

$\frac{2}{x - 1} \cdot \frac{3 \left(x + 1\right)}{3 \left(x + 1\right)} - \frac{4}{x + 1} \cdot \frac{3 \left(x - 1\right)}{3 \left(x - 1\right)} - \frac{2}{3} \cdot \frac{\left(x - 1\right) \left(x + 1\right)}{\left(x - 1\right) \left(x + 1\right)} = 0$

This will be equivalent to

$\frac{6 \left(x + 1\right) - 12 \left(x - 1\right) - 2 \left(x - 1\right) \left(x + 1\right)}{3 \left(x - 1\right) \left(x + 1\right)} = 0$

Next, focus on the numerator. Expand the parentheses and group like terms to get

$6 x + 6 - 12 x + 12 - 2 {x}^{2} + 2 = 0$

$- 2 {x}^{2} - 6 x + 20 = 0$

You can divide all the terms by $- 2$ to get equivalent form

${x}^{2} + 3 x - 10 = 0$

Two numbers that add up to give $3$ and multiply to give $- 10$ are $- 2$ and $5$, which means that the above quadratic equation can be factored as

$\left(x - 2\right) \left(x + 5\right) = 0 \implies \left\{\begin{matrix}{x}_{1} = 2 \\ {x}_{2} = - 5\end{matrix}\right.$

Since

${x}_{1} , {x}_{2} \ne \pm 1$

you can say that the original equation has two possible solutions

$x = 2 \text{ " "or" " } x = - 5$

Do a quick check to make sure that the calculations are correct

$x = 2 : \text{ " 2/(2-1) - 2/3 = 4/(2+1) <=> 2 - 2/3 = 4/3 " } \textcolor{g r e e n}{\sqrt{}}$

$x = - 5 : \text{ " 2/(-5 -2) -2/3 = 4/(-5 + 1) <=> -1/3 - 2/3 = -1 " } \textcolor{g r e e n}{\sqrt{}}$