Finding all r ∈ R such that r<1+1/r?

Find all r ∈ R such that r<1+1/r?

So I'm having difficulty with this problem.

I've figured out that r is a variable and r does not = 0 and by solving it, multiplying both sides by r is incorrect.

Can you guys help me with this problem?

My first instinct is to just multiply both sides by r but that's apparently wrong.

Any help is appreciated.

I got some help elsewhere. Thanks though.

Re: Finding all r ∈ R such that r<1+1/r?

Quote:

Originally Posted by

**paperclip5** Find all r ∈ R such that r<1+1/r?

So I'm having difficulty with this problem.

I've figured out that r is a variable and r does not = 0 and by solving it, multiplying both sides by r is incorrect.

Can you guys help me with this problem?

My first instinct is to just multiply both sides by r but that's apparently wrong.

Any help is appreciated.

I got some help elsewhere. Thanks though.

It's not that multiplying both sides by r is wrong- you just have to be careful. If you multiply both sides of an inequality by a **negative** number, you **reverse** the direction of the inequality.

Yes, as you say, r cannot be 0- so it is either positive or negative. If it is positive, multiplying both sides by r gives $\displaystyle r^2< r+ 1$ which is the same as $\displaystyle r^2- r- 1< 0$. If r is negative, multiplying both sides by r gives $\displaystyle r^2> r+ 1$ which is the same as $\displaystyle r^2- r- 1> 0$. Obviously, determining where $\displaystyle r^2- r- 1= 0$ is important in solving those! Don't forget that the first is only true for r> 0 and the second for r< 0.