Solving a rational equation with multiple and nested fractionsThis is the equation to solve: x + 1 2 1 2 + x 3 1 4 + x 5 = 3What I did: x + 1 2 = 2 x + 1 2 x 3 + 1 2 = 2 x + 3 6 x 5 + 1 4 = 4 x + 5 20 2 x + 1 2 ÷ 2 x + 3 6 = 2 x + 1 2 × 6 2 x + 3 = 6 x + 3 2 x + 3 6 x + 3 2 x + 3 ÷ 4 x + 5 20 = 6 x + 3 2 x + 3 × 20 4 x + 5 = 3 ⟹ 120 x + 60 ( 2 x + 3 ) ( 4 x + 5 ) = 3I know how to solve the equation. But right now I tried several times and I got wrong answers.So I appreciate your help. Thank you.

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Solving a rational equation with multiple and nested fractionsThis is the equation to solve:                                           x            +                                          1                2                                                                                        1                2                                      +                                          x                3                                                                                      1            4                          +                              x            5                                =  3What I did:  x  +            1      2        =                    2        x        +        1            2                  x      3        +            1      2        =                    2        x        +        3            6                  x      5        +            1      4        =                    4        x        +        5            20                          2        x        +        1            2        ÷                    2        x        +        3            6        =                    2        x        +        1            2        ×            6              2        x        +        3              =                    6        x        +        3                    2        x        +        3                                6        x        +        3                    2        x        +        3              ÷                    4        x        +        5            20        =                    6        x        +        3                    2        x        +        3              ×            20              4        x        +        5              =  3    ⟹                      120        x        +        60                    (        2        x        +        3        )        (        4        x        +        5        )              =  3I know how to solve the equation. But right now I tried several times and I got wrong answers.So I appreciate your help. Thank you.

Donavan Mack · Accepted Answer

You have, essentially,            120      x      +      60              (      2      x      +      3      )      (      4      x      +      5      )        =  3Multiply both sides by   (  2  x  +  3  )  (  4  x  +  5  ) to get  120  x  +  60  =  3  (  2  x  +  3  )  (  4  x  +  5  )  =  24      x    2    +  66  x  +  45Subtract   120  x  +  60 from both sides to get  24      x    2    −  54  x  −  15  =  0Divide both sides by 3 to get  8      x    2    −  18  x  −  5  =  0which factors asto yield   x  =  −  1      /    4 or   5      /    2. (Or, you can use the quadratic formula.)

telegrafyx · Accepted Answer

x        +                  1          2                                      1          2                +                  x          3                                    1        4            +              x        5              =  3                    3        (        2        x        +        1        )                    (        2        x        +        3        )                            4        x        +        5            20        =  3            60      (      2      x      +      1      )              (      2      x      +      3      )      (      4      x      +      5      )        =  3  8      x    2    −  18  x  −  5  =  0  x  =            18      ±              (        −        18                  )          2                −        4        (        8        )        (        −        5        )                    2      (      8      )        x  =            18      ±      22        16    x  =      5    2  or  x  =  −      1    4

Solving a rational equation with multiple and nested fractions This is the equation to solve: <

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