How do I simplify and calculate this inequality? log ⁡ ( x 3 ) &gt; | x − 1 | I can't figure out how to go about solving this inequality, besides this one step: 3 log ⁡ ( x ) &gt; | x − 1 |

Question

How do I simplify and calculate this inequality?  log  ⁡  (      x    3    )  &amp;gt;      |    x  −  1      |  I can&#039;t figure out how to go about solving this inequality, besides this one step:  3  log  ⁡  (  x  )  &amp;gt;      |    x  −  1      |

Allison Pena · Accepted Answer

As said in the comments, you have two situations buit you can notice that for   x  =  1, the lhs and rhs are equal.So consider the case where   x  =  1  −  ϵ and use Taylor expansion for the lhs at   x  =  1. Yous will easily see that the inequality is not satisfied. Since you noticed that for   x  =  1,   l  h  s  =  r  h  s, the inequality will be satisfied for   x  &amp;gt;  1. So, you can now forget the absolute value in the rhs.Now, compute the derivative of   l  h  s  −  r  h  s; it is equal to   3      /    x  −  1 and so canceled for   x  =  3; the second derivative being negative, then this point corresponds to a maximum. For   x  =  3,   l  h  s  =  3  l  o  g  (  3  ) and   r  h  s  =  2; so, at this point,   l  h  s  &amp;gt;  r  h  s. On the other side, you know that   x moves faster then   l  o  g  (  x  ); so there is a point which will corresponds to an   x intercept. You will then need to solve the equation  log  ⁡  (      x    3    )  =  x  −  1  log  ⁡  (      x    3    )  =  x  −  1which does not any simple analytical solution. If you plot the function, you will see that the solution is close to   x  =  6.71So, the inequality is satisfied for   1  &amp;lt;  x  &amp;lt;  6.71

How do I simplify and calculate this inequality? log ⁡ ( x

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