Question 117079
{{{2-1/(2+1/((x-1)/x))}}} Start with the given expression



{{{2-1/(2+x/(x-1)))}}} Simplify {{{1/((x-1)/x)}}} to get {{{x/(x-1)}}}. What's really happening is I'm taking the reciprocal of this fraction




{{{2-1/((2)((x-1)/(x-1))+x/(x-1)))}}} Multiply 2 by {{{(x-1)/(x-1)}}}



{{{2-1/((2x-2)/(x-1)+x/(x-1)))}}} Distribute


{{{2-1/((2x-2+x)/(x-1)))}}} Combine the fractions



{{{2-1/((3x-2)/(x-1)))}}} Combine like terms



{{{2-(x-1)/(3x-2)}}} Simplify {{{1/((3x-2)/(x-1))}}} to get {{{(x-1)/(3x-2)}}} by once again taking the reciprocal





{{{2((3x-2)/(3x-2))-(x-1)/(3x-2)}}} Multiply 2 by {{{(3x-2)/(3x-2)}}}



{{{(6x-4)/(3x-2)-(x-1)/(3x-2)}}} Distribute



{{{(6x-4-(x-1))/(3x-2)}}} Combine the fractions




{{{(6x-4-x+1)/(3x-2)}}} Distribute the negative



{{{(5x-3)/(3x-2)}}} Combine like terms




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Answer:


So {{{2-1/(2+1/((x-1)/x))}}} simplifies to {{{(5x-3)/(3x-2)}}}. In other words, {{{2-1/(2+1/((x-1)/x))=(5x-3)/(3x-2)}}}. You can verify this if you graph the original and the simplified expressions and you'll notice that they overlap. So this visually verifies our answer.