My teacher wrote:
1. $\displaystyle \frac{4}{(x+2)^3}>0$
2. $\displaystyle x+2>0$
3. $\displaystyle x>-2$
How can 1. become 2. ???
Is it even a valid move?
I'm confused about where $\displaystyle (x-2)^4$ comes from?
What's happening is my teacher is going from step 1. to step 2. to step 3.
I wanted to know if those moves are algebraically correct because I'm not quite sure on inequalities and I'm worried that going from step 1. to step 2. might be wrong. Is it wrong?
I think this might be what's happening:
$\displaystyle \frac{4}{(x+2)^3}>0$
$\displaystyle \frac{4}{(x+2)^3}*(x+2)^4>0*(x+2)^4$
$\displaystyle 4(x+2)>0$
$\displaystyle (x+2)>0$
$\displaystyle x>-2$
Is this correct?
Actully $\displaystyle (x-2)^4$ does not come into it.
It is a logical solution not operational.
If we know that $\displaystyle \frac{4}{a^3}>0$ then we know at once $\displaystyle a>0$.
It can be no other way.
If $\displaystyle a<0$ then $\displaystyle \frac{4}{a^3}>0$ is false..
So your teacher just knew that $\displaystyle (x-2)>0$ from the logic of inequalities.