Question

How do you solve `x^3<9x`?

Answer 1

`color(green)(x in (0,3))` or `color(green)(x in (-oo,-3))`

Explanation:

Given
`color(white)("XXX")x^3 < 9x`

Note that the equation can not be true if `x=0`

Case 1: `color(black)(x > 0)`
`color(white)("XXX")x^2 < 9`
`color(white)("XXX")rarr x < 3`
therefore `x in (0,3)`

Case 2: `color(black)(x < 0)`
`color(white)("XXX")x^2 > 9` (remember when dividing by a negative the inequality is reversed)
`color(white)("XXX")abs(x) > 3`
and since `x < 0`
`color(white)("XXX")rarr x < -3`
therefore `x in (-oo,-3)`