If x ≥ − 1, prove that ( x + 1 ) 3 = x has no solutions

Question

If   x  ≥  −  1, prove that   (  x  +  1      )    3    =  x has no solutions

Cristopher Barrera · Accepted Answer

If   x  ≥  −  1 then   x  +  1  ≥  0. When you cube this, it will thus be positive. This can only equal x again if   x  &amp;gt;  0. But if   x  &amp;gt;  0, it is easy to see that   (  x  +  1      )    3    =  x is impossible.

Abram Boyd · Accepted Answer

The given equation is  x  (  x  +  1  )  (  x  +  2  )  =  −  1Now   x  ≥  −  1If   x  ≥  0, then RHS cannot be equal to LHS as positive or zero is never equal to negative.If   x  =  −  1 , then LHS=0 but RHS is not, therefore equality doesn&#039;t holdsNow we are left to check for   x  ∈  (  −  1  ,  0  ). For that we use our original equation  (  x  +  1      )    3    =  x. If   x  ∈  (  −  1  ,  0  ), then   x  +  1  ∈  (  0  ,  1  ) and therefore   (  x  +  1      )    3    &amp;gt;  0But then LHS becomes positive and RHS remains negative, therefore again equality doesn&#039;t holdsHence proved.

If x ≥ −<!--

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