Question

How do you solve and write the following in interval notation: `x^3 - 38x^2 <0`?

Answer 1

Solution is `x<38`, except `x=0`

Explanation:

`x^3-38x^2<0` can be factorized as

`x^2(x-38)<0`

Now, if `x<0`, `x^2(x-38)<0` as `x^2` is positive, but `(x-38)` is negative and their product too is negative and inequality is satisfied.

If `0 < x < 38`, while `x^2` is positive, `(x-38)` is negative and their product too is negative and inequality is satisfied.

But if `x>38`, `x^2` and `(x-38)` are positive, and their product too is positive, and inequality is not satisfied.

Hence, solution is `x<38`, but `x=0` is not one as at `x=0`, `x^3-38x^2<0` is not true.