If x is positive, then why does 1 x + 1 + x = x + 1 − x ?I've been trying to convert the left side of the equation to the right side: 1 x + 1 + x But then how can I flip this round to be what I have on the right side?I know that 1 x = x − 1 2 , so I would think that this would give me ( x + 1 ) − 1 / 2 + x − 1 / 2 Which I thought would then convert to − x + 1 + ( − x )so I'm not sure how the first part x + 1 got to be positive

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If   x is positive, then why does       1                  x        +        1            +              x              =      x    +    1    −      x  ?I&#039;ve been trying to convert the left side of the equation to the right side:      1                  x        +        1            +              x            But then how can I flip this round to be what I have on the right side?I know that       1          x        =      x          −              1        2            , so I would think that this would give me  (  x  +  1      )          −      1              /            2        +      x          −      1              /            2      Which I thought would then convert to  −      x    +    1    +  (  −      x    )so I&#039;m not sure how the first part       x    +    1   got to be positive

odczepneyv · Accepted Answer

Rationalize the denominator:                              1                                    x              +              1                        +                          x                                      ⋅                                            x              +              1                        −                          x                                                          x              +              1                        −                          x                                                          =                                            x              +              1                        −                          x                                                                                                                (                                          x                      +                      1                                        +                                          x                                        )                                    ⋅                                      (                                          x                      +                      1                                        −                                          x                                        )                                                  ⏟                                                                    FOIL                                                                                      =        ⋯            Some notes:If you want to show that two things are equal to each other, it&#039;s better to manipulate only one of them than it is to start by saying they&#039;re equal to each other and then manipulating both sides to verify, etc.Your idea of converting to   (  x  +  1      )          −      1              /            2        +      x          −      1              /            2       won&#039;t work because, in general,             1              a        +        b              ≠          1      a        +          1      b

Bailee Landry · Accepted Answer

Clear the denominator to get:  1  =  (      x    +    1    +      x    )  (      x    +    1    −      x    )  =  (  x  +  1  )  −  x  .As Tilper points out, this is not a correct proof per se, but once you see this, you can reverse the ordering of your logic and write a proper proof.In this case it would be more proper to write:  1  =  (  x  +  1  )  −  x  =  (      x    +    1    +      x    )  (      x    +    1    −      x    )  .Therefore since       x    +    1    +      x    ≠  0      1                  x        +        1            +              x              =      x    +    1    −      x    .

If x is positive, then why does 1 <msqrt> x +

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