# Solve for x: 1+1/(1+(1/(1+1/x))=4?

Oct 17, 2017

$x = - \frac{2}{5}$ or $- 0.4$

#### Explanation:

Move $1$ to the right hand side of the equation so you get rid of it.

1/(1+(1)/((1+1/x))$= 4 - 1$

1/(1+(1)/((1+1/x))$= 3$

Then, multiply both sides by the denominator $1 + \frac{1}{1 + \left(\frac{1}{x}\right)}$ so that you can cancel it out.

1/cancel((1+(1)/((1+1/x)))$= 3 \left(1 + \frac{1}{1 + \left(\frac{1}{x}\right)}\right)$

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

Move $3$ to the left hand side.

-2=3/(1+(1/x)

Again, multiply by the denominator so you can cancel it out.

-2(1+1/x)=3/cancel(1+(1/x)

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

Solve for $x$.

$- \frac{2}{x} = 5$

$x = - \frac{2}{5}$ or $- 0.4$

To check if the answer is correct, substitute the $x = - \frac{2}{5}$ into the equation. It gives you $4$.

Oct 17, 2017

$x = - \frac{2}{5}$

#### Explanation:

Note that provided an equation is non-zero, then taking the reciprocal of both sides results in an equation which holds if and only if the original equation holds.

So one method of addressing the given example goes as folows..

Given:

$1 + \frac{1}{1 + \left(\frac{1}{1 + \frac{1}{x}}\right)} = 4$

Subtract $1$ from both sides to get:

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

Take the reciprocal of both sides to get:

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

Subtract $1$ from both sides to get:

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

Take the reciprocal of both sides to get:

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

Subtract $1$ from both sides to get:

$\frac{1}{x} = - \frac{5}{2}$

Take the reciprocal of both sides to get:

$x = - \frac{2}{5}$

Since all of the above steps are reversible, this is the solution of the given equation.