How do I find x in (2/x) - (2/2x+1) = (1/x+1)?

Question

I've tried making the denominator the same, which ended up in polynomials. I think I did the wrong steps.

Answer 1

Restrict the domain: x ∉ {-0.5,  -1, 0}

 

2/x - 2/(2x + 1) = 1/(x + 1)                      multiply both sides by LCM

2(2x + 1)(x + 1) - 2x(x + 1) = x(2x + 1)   distribute

(4x + 2)(x + 1) - 2x² - 2x = 2x² + x         FOIL

4x² +6x + 2 - 2x² - 2x = 2x² + x             combine like terms

2x² + 4x + 2 = 2x² + x                            subtract 2x² from both sides

4x + 2 = x                                                subtract 2 from both sides

4x = x - 2                                                subtract x from both sides

3x = -2                                                    divide both sides by 3

x = -2/3

 

Answer 2

I also get x =-2/3

Answer 3

Let's start by looking at one part of the problem:  2/2x.  Since 2 is in the numerator and denominator, it cancels to leave 1/x.  So, the problem becomes

 

(2/x) - (1/x + 1) = (1/x + 1). 

 

Our solution can't have x in the denominator.  So a good place to go from here is to multiply both sides of the equation by x:

 

x [ (2/x) - (1/x + 1) ] = x [ (1/x + 1) ] .

 

We can then use the distributive property, x(a+b) = ax + bx, or x(a-b) = ax - bx.  We distribute the x on both sides of the equation:

 

x(2/x) - x(1/x + 1) = x(1/x) + x(1).  We have to use the distributive property again on the left hand side of the equation.  Be careful with the signs, since we're distributing -x, not x:

 

x(2/x) - x(1/x) - x(1) = x(1/x) + x(1).  We carry out the multiplications by x:

 

2 - 1 - x = 1 + x.  Find that 2-1 = 1:

1 - x = 1 + x.  Add x to both sides:

1 = 1 + 2x.   Subtract 1 from both sides:

0 - 2x.  Divide both sides by 2:

0 = x.

 

Let's think about this for a sec; 0 seems like the right solution.  We used algebra perfectly to find it.  But if we look at the original equation, we see that 0 would be in the denominator of each fraction!  This means that there is no solution.  0 is the only number which works, but it can't be used since each fraction would be a division by 0.

 

The final answer is there is no solution.

Answer 4

Get the first denominator the same as the second one by multiplying the first part ( 2/x ) by ( (2+(1/x)) / (2+(1/x)) ) which is the same as multiplying it by 1.  That gives you:  (2 (2+(1/x))) / (x (2+(1/x)) )

 

That simplifies to  (4 + (2/x)) / (2x + 1)

Now the first and second parts have the same denominator and you can add/subtract them together.  By this point you should have:

 

( 2 + (2/x) ) / (2x+1)  =  1 / (x+1)

 

Now you can cross multiply giving you:

(x+1)(2+(2/x)) = 2x + 1

 

Multiplying out the first two parts gives you:

2x + 2 + 2 + 2/x = 2x + 1

 

Reducing to:

4 + 2/x = 1

 

Multiplying by X to both sides:

4x + 2 = x

3x = -2

x = -2/3