# Inequalities with a variable in the denominator

• Jul 6th 2009, 12:36 PM
Chokfull
Inequalities with a variable in the denominator
I'm past this, but couldn't remember the rule....I have
$\displaystyle \frac {5} {(x+1)^3} < N$
Then there is a rule for solving this.... You can't multiply both sides by$\displaystyle (x+1)^3$,because you don't know if $\displaystyle x$ is negative or positive. The book says you should get
$\displaystyle {\sqrt [3]{5/|N|}}$
Thanks!
• Jul 6th 2009, 01:52 PM
craig
Quote:

Originally Posted by Chokfull
I'm past this, but couldn't remember the rule....I have
$\displaystyle \frac {5} {(x+1)^3} < N$
Then there is a rule for solving this.... You can't multiply both sides by$\displaystyle (x+1)^3$,because you don't know if $\displaystyle x$ is negative or positive. The book says you should get
$\displaystyle {\sqrt [3]{5/|N|}}$
Thanks!

With these types of equalities you have a couple of chouces.

As you said you cannot multiply both sides by$\displaystyle (x+1)^3$, however you can multiply both by$\displaystyle (x+1)^6$, as you know this is a positive number.

You can also make both sides into fractions and add or subtract to rearrange.

For example,$\displaystyle \frac {5} {(x+1)^3} <\frac{N(x+1)^3}{(x+1)^3}$, leading to $\displaystyle \frac {5} {(x+1)^3} - \frac{N(x+1)^3}{(x+1)^3}<0$

Can you solve this from here?
• Jul 9th 2009, 11:57 AM
Chokfull
Quote:

Originally Posted by craig
With these types of equalities you have a couple of chouces.

As you said you cannot multiply both sides by$\displaystyle (x+1)^3$, however you can multiply both by$\displaystyle (x+1)^6$, as you know this is a positive number.

You can also make both sides into fractions and add or subtract to rearrange.

For example,$\displaystyle \frac {5} {(x+1)^3} <\frac{N(x+1)^3}{(x+1)^3}$, leading to $\displaystyle \frac {5} {(x+1)^3} - \frac{N(x+1)^3}{(x+1)^3}<0$

Can you solve this from here?

OK, but how does that get an absolute value? I'll fiddle with it for now but don't see how it gets $\displaystyle {\sqrt [3]{5/|N|}}$