Hi. I had the problem below for homework. I think it is true, but I don't know why. Can anyone give me a reason why this is true, because I need to provide an explination. Thanks!

True or False:

If a<b then a3 <b3

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- Sep 9th 2009, 01:57 PMsleighHelp with True/False algebra inequality
Hi. I had the problem below for homework. I think it is true, but I don't know why. Can anyone give me a reason why this is true, because I need to provide an explination. Thanks!

True or False:

If a<b then a3 <b3

- Sep 9th 2009, 02:07 PM11rdc11
- Sep 9th 2009, 02:16 PMPlato
- Sep 9th 2009, 02:17 PMMatt Westwood
I assume you mean: if $\displaystyle a < b$ then $\displaystyle a^3 < b^3$.

It is actually true, but needs a fair amount of work to demonstrate.

First, you can show it's true when both $\displaystyle a$ and $\displaystyle b$ are positive, as follows.

You know that if $\displaystyle a, b, c < 0$ then if $\displaystyle a < b$ then $\displaystyle a c < b c$.

Okay, so $\displaystyle a < b$ implies $\displaystyle a^2 < a b$ (putting $\displaystyle c = a$ in the above).

Then $\displaystyle a < b$ implies $\displaystyle ab < b^2$ (putting $\displaystyle c = b$ in the above).

So you've shown $\displaystyle a < b$ implies $\displaystyle a^2 < b^2$.

Do the same (similar) thing to prove for $\displaystyle a^3 < b^3$.

You go through a similar process when $\displaystyle a$ and $\displaystyle b$ are both negative, and again when $\displaystyle a$ is negative and $\displaystyle b$ positive.

An alternative approach makes use of the fact that the cube function is monotonically increasing - but that's really on a calculus course. - Sep 9th 2009, 03:57 PM11rdc11
- Sep 9th 2009, 07:56 PMMozart
$\displaystyle a < b = (a<b)^3 = \boxed{a^3 < b^3}$

- Sep 9th 2009, 10:09 PMMatt Westwood
- Sep 9th 2009, 11:48 PMLogic
I trust this works?

- Sep 10th 2009, 01:00 PMMatt Westwood
How sure are we that $\displaystyle a^2 + ab + b^2 > 0$?

- Sep 10th 2009, 04:11 PMKrizalid
$\displaystyle a^{2}+ab+b^{2}=\frac{4a^{2}+4ab+4b^{2}}{4}=\frac{( 2a+b)^{2}+3b^{2}}{4}>0.$

Though you can't manipulate what you want to prove.