# Help solving for $x$ in the equation $x= \frac{1}{1+1/x}$

The question asks if it is possible to solve for $x$, and if it is, what is $x$?

$$x= \frac{1}{1+1/x}$$

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That was my first answer, but my teacher said it was wrong. Maybe it's supposed to be x = 1 divided by (1+1)/x? –  samm_hall Feb 4 '13 at 6:48

Suppose that there is a real number $x$ satisfying the equation.

Multiply both sides of the equation by $1+1/x$. After some simplification, we get $x+1=1$, so $x=0$. But this is ridiculous, $1/x$ is not defined at $0$. Our supposition was wrong. There is no solution!

Remark: This kind of thing happens moderately often. When we are solving an equation, we sometimes do complicated manipulations. Some of these manipulations can produce fake (the technical term is extraneous) roots. So when you have produced what you think is a root or roots to an equation, it is very useful to substitute the numbers obtained into the equation, to see whether they "work." This also serves to detect errors of calculation.

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This turned out to be the correct answer, thank you. –  samm_hall Feb 4 '13 at 6:55

Well we can just multiply both sides by $1+\frac1x$ to get

$x\big(1+\frac1x\big)=1$

then

$x+1=1$

$x=0$

But $x$ can't be $0$ , since this would leave the fraction on the right side undefined.

I don't think there are any complex values of $x$ that satisfy the equation either. So no, it's not possible to solve this equation.

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At no point in your answer did you assume $x$ was real, so your argument shows there are no complex solutions either. –  Michael Albanese Feb 4 '13 at 7:38