Showing this fraction is less than one?And also, what are some techniques to show such a thing. That is, what are standard techniques for showing a fraction is less than one? I'm just trying to refresh/improve/add to my toolbox.The fraction I am working with is: t x ( t x + 1 ) + t 2 ( t x + 1 ) 2 The problem is: Show t x ( t x + 1 ) + t 2 ( t x + 1 ) 2 ≤ 1 for x ≥ t , t ≥ 0I have been trying to move the denominator to the right and show that the numerator is less than the denominator, but I am not seeing any clean cancellations, or obvious result.

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Showing this fraction is less than one?And also, what are some techniques to show such a thing. That is, what are standard techniques for showing a fraction is less than one? I&#039;m just trying to refresh/improve/add to my toolbox.The fraction I am working with is:            t      x      (      t      x      +      1      )      +              t        2                    (      t      x      +      1              )        2            The problem is: Show            t      x      (      t      x      +      1      )      +              t        2                    (      t      x      +      1              )        2              ≤  1   for   x  ≥  t  ,     t  ≥  0I have been trying to move the denominator to the right and show that the numerator is less than the denominator, but I am not seeing any clean cancellations, or obvious result.

Nola Rivera · Accepted Answer

Since   x  ≥  t  ≥  0, we have   t  x  ≠  −  1 and so                              t          2                ≤        t        x                            ⇒                          t          2                ≤        t        x        +        1                                  ⇒                t        x        (        t        x        +        1        )        +                  t          2                ≤        t        x        (        t        x        +        1        )        +        t        x        +        1                                  ⇒                t        x        (        t        x        +        1        )        +                  t          2                ≤        (        t        x        +        1                  )          2                         .

Showing this fraction is less than one? And also, what are some techniques to show such a thing. Th

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