Finding Range for r in Inequality: Quick Question on Solving for Lower Bound

[azdang]

I first accidentally posted this in Calculus & Beyond since it is for a 700-level class, but I'm realizing now that it's pretty basic, and it should probably go here:

I'm working on a problem in which I have to find a range for r. I have an upper bound on it, but I can't seem to get the lower bound.

Here is the inequality to start with:
$$\lambda$$r - r³ + $$\lambda$$ < 0

Eventually, I get it down to:
$$\lambda$$ < $$\frac{r^3}{r+1}$$

However, I need r by itself on one side, and I have no idea what to do. Is there anything I actually could do or am I stuck?

Another note: r>0 and $$\lambda$$>0. Thanks!

 

[HallsofIvy]

I first accidentally posted this in Calculus & Beyond since it is for a 700-level class, but I'm realizing now that it's pretty basic, and it should probably go here:

I'm working on a problem in which I have to find a range for r. I have an upper bound on it, but I can't seem to get the lower bound.

Here is the inequality to start with:
$$\lambda$$r - r³ + $$\lambda$$ < 0

Eventually, I get it down to:
$$\lambda$$ < $$\frac{r^3}{r+1}$$

However, I need r by itself on one side, and I have no idea what to do. Is there anything I actually could do or am I stuck?

Another note: r>0 and $$\lambda$$>0. Thanks!

You can't in any simple way. You can use the fact that points where one side is equal separate "<" from ">". However you still need to solve $r^3- \lamba r- \lambda= 0$. There is a "cubic formula" but it is very complicated.
http://www.math.vanderbilt.edu/~schectex/courses/cubic/

 

[azdang]

Okay, yeah, I've worked with the cubic formula in undergrad, but I very highly doubt we're expected to be using it here. I'll have to check with my teacher tomorrow, thanks so much!