1. ## Inequality Question

What's the best method for solving these without a graph.

Find $\displaystyle \left\{ x:x^{\frac{3}{2}}>x^2 \right\}$

2. Originally Posted by Stroodle
What's the best method for solving these without a graph.
Find $\displaystyle \left\{ x:x^{\frac{3}{2}}>x^2 \right\}$
It should be clear to you that $\displaystyle x\ge 0$ else $\displaystyle x^{\frac{3}{2}}$ is not defined.
Therefore $\displaystyle x^3>x^4$.
Now go for it.

3. Originally Posted by Stroodle
What's the best method for solving these without a graph.

Find $\displaystyle \left\{ x:x^{\frac{3}{2}}>x^2 \right\}$

$\displaystyle x^{\frac{3}{2}}-x^2>0$

$\displaystyle x^{\frac{3}{2}}(1-x^{\frac{1}{2}})>0$

Draw the number line .
THus the solution set is {$\displaystyle {x: 0<x<1 }$}

Definitely not the best solution .

4. It's clear that both sides are positive, so you can square them, thus $\displaystyle x^3>x^4\implies x^2\cdot x(x-1)<0.$

I think you can continue.