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) About Me  (Show Source):
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
1%2F%281%2B1%2F%281-1%2Fx%29%29=2


Start with the simplest rational sub-expression you can find, and work outward.
1%2F%281%2B1%2F%28%28x-1%29%2Fx%29%29=2

1%2F%28%28x%2Bx-1%29%2Fx%29=2

1%2F%28%282x-1%29%2Fx%29=2

1%2Ax%2F%28%28%282x-1%29%2Fx%29x%29=2

x%2F%282x-1%29=2
Much less complicated now.

x=2%282x-1%29

x=4x-2

2=3x

x=2%2F3

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

 

1 ÷ matrix%281%2C3%2C+%281+%2B+x%2F%28x+-+1%29%29%2C+%22=%22%2C+2%29
1 ÷ matrix%281%2C3%2C+%28%28x+-+1+%2B+x%29%2F%28x+-+1%29%29%2C+%22=%22%2C+2%29
  1 ÷ matrix%281%2C3%2C+%28%282x+-+1%29%2F%28x+-+1%29%29%2C+%22=%22%2C+2%29
 matrix%281%2C5%2C+1%2C+%22%2A%22%2C+%28%28x+-+1%29%2F%282x+-+1%29%29%2C+%22=%22%2C+2%29
        matrix%281%2C3%2C+%28x+-+1%29%2F%282x+-+1%29%2C+%22=%22%2C+2%29
          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) About Me  (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
1%2F%281%2B1%2F%28%28x-1%29%2Fx%29%29=2
to
1%2F%28%28x%2Bx-1%29%2Fx%29=2
(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) About Me  (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.

1-1%2Fx=x%2Fx-1%2Fx=%28x-1%29%2Fx

1%2F%28%28x-1%29%2Fx%29=x%2F%28x-1%29

1%2Bx%2F%28x-1%29=%28x-1%29%2F%28x-1%29%2Bx%2F%28x-1%29=%282x-1%29%2F%28x-1%29

1%2F%28%282x-1%29%2F%28x-1%29%29=%28x-1%29%2F%282x-1%29

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

%28x-1%29%2F%282x-1%29=2
2%282x-1%29=x-1
4x-2=x-1
3x=1
x=1%2F3

ANSWER: 1/3