Is 1 ( x + 3 ) 2 strictly greater than 2 x - 3 ?

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Is             1                        (          x          +          3          )                2             strictly greater than             2              x        -        3            ?

luisjoseblash2 · Accepted Answer

Step 1If it was, then the answer to &quot;For which values of x the following disequality holds?&quot;            1                        (          x          +          3          )                2              -          2              x        -        3              &amp;gt;    0  Should be &quot;Always&quot;. Let&#039;s check it. Make the fractions have a common denominator, and you&#039;ll end up with                     x        -        3                                          (            x            +            3            )                    2                          (          x          -          3          )                      -                  2                              (            x            +            3            )                    2                                                  (            x            +            3            )                    2                          (          x          -          3          )                      &amp;gt;    0  Step 2Now sum the numerator:            (      x      -      3      )        -    2                  (        x        +        3        )            2        =    x    -    3    -    2          (              x        2            +      6      x      +      9      )        =        =    x    -    3    -    2          x      2        -    12    x    -    18    =        =    -    2          x      2        -    11    x    -    21    =    -          (      2              x        2            +      11      x      +      21      )      Step 3SInce       2          x      2        +    11    x    +    21   has a negative discriminant, it has no solution, and so it is always positive. So, the fraction becomes       -                  2                  x          2                +        11        x        +        21                                          (            x            +            3            )                    2                          (          x          -          3          )                    .and we know that       2          x      2        +    11    x    +    21   and                     (        x        +        3        )            2       are always positive (the latter being a square). So, the sign of the whole fraction depends on the sign of       x    -    3  , which obviously is positive after 3 and negative before. Since there is a minus before the whole fraction, we have that the expression is positive only before 3, and so the answer is no:            1                        (          x          +          3          )                2             is not always greater than             2              x        -        3            , but only before 3.

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