# How do you solve \frac { 1} { 5- \frac { 1} { 1- \frac { 1} { x } } } = \frac { 2} { 7}?

Dec 7, 2017

$x = 3$

#### Explanation:

This is kind of a mess, but as long as we take it step by step, we'll be fine

$\frac{1}{5 - \frac{1}{1 - \frac{1}{x}}} = \frac{2}{7}$

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Let's make $1 - \frac{1}{x}$ one fraction with a common denominator

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$\frac{1}{5 - \frac{1}{\frac{x - 1}{x}}} = \frac{2}{7}$

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Now, let's change this mess of $\frac{1}{\frac{x - 1}{x}}$ to $\frac{x}{x - 1}$

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$\frac{1}{5 - \frac{x}{x - 1}} = \frac{2}{7}$

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Now, if we change $5$ to $\frac{5}{1}$, we can give it the same denominator: $\frac{5 x - 5}{x - 1}$

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That gives us

$\frac{1}{\frac{5 x - 5 - x}{x - 1}} = \frac{2}{7}$

or

$\frac{x - 1}{4 x - 5} = \frac{2}{7}$

See how much better this looks!! We are almost there

Let's clear the denominator

$7 x - 7 = 8 x - 10$

$x = 3$

There! We are done, good work