SOLUTION: 1/[1+1/(1-1/x)]=2

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Question 1055130: 1/[1+1/(1-1/x)]=2
Found 5 solutions by ikleyn, josgarithmetic, MathTherapy, math_tutor2020, greenestamps:
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.
x = .


Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!



Start with the simplest rational sub-expression you can find, and work outward.









Much less complicated now.








Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
1/[1+1/(1-1/x)]=2

 

1 ÷ 
1 ÷ 
  1 ÷ 
 
        
          x - 1 = 2(2x - 1) --- Cross-multiplying, or multiplying by LCD, 2x - 1
          x - 1 = 4x - 2
         x - 4x = - 2 + 1
           - 3x = - 1
             

As usual, the person named JOGS/JOSG is AGAIN INCORRECT, as he normally is. Therefore, ONCE AGAIN,
IGNORE his solution, if it can be called that!
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

I apologize if my response is a bit late compared to the other tutors.
The answer is x = 1/3 and not x = 2/3.

The tutor josgarithmetic made an error when going from

to

(steps 2 and 3 of his/her scratch work).

------------------------------------

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
x = 1/3 which is the final answer.

You can verify using WolframAlpha

GeoGebra is another tool you can use to verify. Use the Solve 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 x = 1/3 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.

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Solve by simplifying the expression on the left using basic rules of algebra, starting with the innermost parentheses and working out.









Now we are ready to write and solve the equation using this simplified form of the expression.







ANSWER: 1/3
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