Simplifying inequality by factoring?

[yo4561]

How would you simplify this statement (given as part an MP practice problem by an instructor):

v^4 <v

Can you not do:
v (v^3 - 1) < 0

because you do not know the sign of v?

Many thanks in advance

[esledge]

You can indeed move all the v terms to the left side and factor a v out, as you showed. And you are exactly right about why you can't divide the whole inequality by v to "get rid of it." (what if v is negative? the sign would flip)

So on this example, you have reached point where you have 0 on one side and you have simplified as far as you can. At that point, you can start thinking about "signs" and basically solve this as a number properties question (not algebra) from here on:
v^4 <v
v (v^3 - 1) < 0

(I would sketch this thinking on a number line, rather than writing it out on the test. Sketch the number line and show the following visually with + and - signs on different segments of the number line.)

The v term is positive when v>0 and negative when v<0.
The v^3-1 term is positive when v>1 and negative when v<1.

Now look at how these combine:
For v<0, both terms are negative and the product is positive.
For 0<v<1, the terms have opposite signs and the product is negative.
For 1<v, both terms are positive and the product is positive.

The only solutions are 0<v<1, then.