1. ## Nonlinear inequality

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set.

I am stuck on where the line should start, I know it is a positive infinity, it appears as if it starts at -1, but its not correct.

2. Start by reasoning on the Domain and Range.

Domain doesn't help much.

Range?

$\displaystyle x^{4} \ge 0$

$\displaystyle x^{9}$ has the same sign as x.

This restricts your solutions to $\displaystyle x > 0$.

After that:

$\displaystyle x^{9} - x^{4} > 0$

$\displaystyle x^{4} \cdot (x^{5} - 1) > 0$

Only $\displaystyle x \ne 0$ for $\displaystyle x^{4}$. How about the other piece?

$\displaystyle x^{5} - 1 = (x-1)(x^{4}+x^{3}+x^{2}+x+1)$

That big piece restricts nothing, being always greater than zero (0).

The only piece of any remaining significance is (x - 1).

This is a great problem to see if you were paying attention in class. Were you?

3. Edit: sorry, didnt see tkhunny's post. Feel free to ignore this
$\displaystyle x^9>x^4$
if $\displaystyle x \not = 0$
$\displaystyle \frac{x^9}{x^4} >1$
$\displaystyle x^5>1$
$\displaystyle (x^5)^{1/5} >1^{1/5}$
$\displaystyle x>1$

if x = 0 then $\displaystyle 0^9\not > 0^4$

I know it is a positive infinity,
Right

4. Well it's like having $\displaystyle x^3>1$ then the cubic root "does the work," since this is just $\displaystyle x>1.$ But this is actually the answer, but it does require a little bit of justification.