Question 1055130
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I apologize if my response is a bit late compared to the other tutors. 
The <font color=red>answer is x = 1/3</font> and not x = 2/3. 


The tutor josgarithmetic made an error when going from 
{{{1/(1+1/((x-1)/x))=2}}}
to
{{{1/((x+x-1)/x)=2}}}
(steps 2 and 3 of his/her scratch work).


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Here's how I would solve 


Let,
p = 1 - (1/x)
q = 1 + (1/p)
These helper equations are created to clean up the messy nested fractions. 
The equation your teacher gave you can be rewritten as 1/q = 2
That solves to q = 1/2


Let's determine p based on this.
q = 1/2
1 + (1/p) = 1/2
2p + 2 = p ......... multiply every term by the LCD 2p to clear out the fractions
2p-p = -2
p = -2


Now we can finally solve for x.
p = -2
p = 1 - (1/x)
1 - (1/x) = -2
x - 1 = -2x ......... multiply every term by the LCD x to clear out the fraction
x+2x = 1
3x = 1
<font color=red>x = 1/3</font> which is the final answer.


You can verify using <a href="https://www.wolframalpha.com/input?i=1%2F%5B1%2B1%2F%281-1%2Fx%29%5D%3D2">WolframAlpha</a>


<a href="https://www.geogebra.org/calculator">GeoGebra</a> is another tool you can use to verify. Use the <a href="https://geogebra.github.io/docs/manual/en/commands/Solve/">Solve</a> command. 
Make sure that the square brackets in 1/[1+1/(1-1/x)]=2 are changed to parenthesis when working with GeoGebra. Otherwise it will result in an error.


Or you can verify by plugging <font color=red>x = 1/3</font> into the original equation and simplifying. Start on the inner most portion and work your way outward.
That means you'll evaluate p = 1 - (1/x) first. Then evaluate q = 1 + (1/p). Then finally evaluate 1/q and you should get 2. 
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