document.write( "Question 1055130: 1/[1+1/(1-1/x)]=2 \n" ); document.write( "
Algebra.Com's Answer #846772 by math_tutor2020(3817)\"\" \"About 
You can put this solution on YOUR website!

\n" ); document.write( "I apologize if my response is a bit late compared to the other tutors.
\n" ); document.write( "The answer is x = 1/3 and not x = 2/3. \r
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\n" ); document.write( "\n" ); document.write( "The tutor josgarithmetic made an error when going from
\n" ); document.write( "\"1%2F%281%2B1%2F%28%28x-1%29%2Fx%29%29=2\"
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\n" ); document.write( "\"1%2F%28%28x%2Bx-1%29%2Fx%29=2\"
\n" ); document.write( "(steps 2 and 3 of his/her scratch work).\r
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\n" ); document.write( "\n" ); document.write( "Here's how I would solve \r
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\n" ); document.write( "\n" ); document.write( "Let,
\n" ); document.write( "p = 1 - (1/x)
\n" ); document.write( "q = 1 + (1/p)
\n" ); document.write( "These helper equations are created to clean up the messy nested fractions.
\n" ); document.write( "The equation your teacher gave you can be rewritten as 1/q = 2
\n" ); document.write( "That solves to q = 1/2\r
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\n" ); document.write( "\n" ); document.write( "Let's determine p based on this.
\n" ); document.write( "q = 1/2
\n" ); document.write( "1 + (1/p) = 1/2
\n" ); document.write( "2p + 2 = p ......... multiply every term by the LCD 2p to clear out the fractions
\n" ); document.write( "2p-p = -2
\n" ); document.write( "p = -2\r
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\n" ); document.write( "\n" ); document.write( "Now we can finally solve for x.
\n" ); document.write( "p = -2
\n" ); document.write( "p = 1 - (1/x)
\n" ); document.write( "1 - (1/x) = -2
\n" ); document.write( "x - 1 = -2x ......... multiply every term by the LCD x to clear out the fraction
\n" ); document.write( "x+2x = 1
\n" ); document.write( "3x = 1
\n" ); document.write( "x = 1/3 which is the final answer.\r
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\n" ); document.write( "\n" ); document.write( "You can verify using WolframAlpha\r
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\n" ); document.write( "\n" ); document.write( "GeoGebra is another tool you can use to verify. Use the Solve command.
\n" ); document.write( "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.\r
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\n" ); document.write( "\n" ); document.write( "Or you can verify by plugging x = 1/3 into the original equation and simplifying. Start on the inner most portion and work your way outward.
\n" ); document.write( "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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