sayak636 wrote:

What is the best way to tackle a question of the following type?

Find the range of values of X if X^3< X^(1/3) and X^2> X^(1/3). I am trying to solve this using algebra.

I prefer to get rid of the fractional exponents, therefore I use substitutions.

For the first inequality:

Denote \(x^{\frac{1}{3}}=y,\) so \(x=y^3.\) Then we have to find the values of \(y\) for which \(y^9<y\) or \(y(y^8-1)<0.\)

You have to check for values of \(y\) such that \(y<-1,\) \(\, -1<y<0,\) \(\, 0<y<1\) and \(y>1.\)

It follows that \(y<-1\) or \(0<y<1\). Translating back to \(x,\) we deduce that \(x<-1\) or \(0<x<1.\)

For the second inequality:

Denote \(x^{\frac{1}{3}}=y\), again \(x=y^3\). Then we have to find the values of \(y\) for which \(y^6>y\) or \(y(y^5-1)>0.\)

Now you have to test for values of \(y\) such that \(y<0,\) \(\, 0<y<1\) and \(y>1.\)

You find that \(y<0\) or \(y>1\). In terms of \(x\), this means \(x<0\) or \(x>1.\)

Look for most posts about inequalities on the site. Here is just one useful one:

solving-inequalities-134671.html
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PhD in Applied Mathematics

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