# Given f(x)=x/(x-1), how do you find the values of x for which f(x)=2/3?

Dec 5, 2016

$x = - 2$

#### Explanation:

First, we must write this problem as"

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

To solve this we should first multiply this equation by a common denominator to eliminate the fractions and to keep the equation balanced. In this case we can multiply by $3 \left(x - 1\right)$:

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

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

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

Next, we can expand the term in parenthesis:

$3 x = 2 x - 2$

Now we can isolate the $x$ term on one side of the equation and the constant on the other side of the equation while keeping the equation balanced:

$3 x - 2 x = 2 x - 2 x - 2$

$3 x - 2 x = 0 - 2$

Now we can consolidate likes terms to solve for $x$:

$\left(3 - 2\right) x = - 2$

$1 x = - 2$

$x = - 2$