Question

How do you solve `x^3-2x^2<5x-6`?

Answer 1

6>-6

Explanation:

x^3-2x^2< 5x-6
-2x^1< 5x-6
-7x^1< -6
x>6/7
7*6/7=6
x=6

Answer 2

The solution is `x in (-oo,-2)uu(1,3)`

Explanation:

The inequality is

`x^3-2x^2<5x-6`

`x^3-2x^2-5x+6<0`

Let `f(x)=x^3-2x^2-5x+6`

Then,

`f(1)=1-2-5-6=0`

`(x-1)` is a factor

Perform a long division

`f(x)=(x-1)(x^2-x-6)=(x-1)(x+2)(x-3)`

Now, build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaa)` `1` `color(white)(aaaaa)` `3` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+2` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-1` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-3` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(x)<0` when `x in (-oo,-2)uu(1,3)`

graph{x^3-2x^2-5x+6 [-14.36, 14.12, -5.01, 9.23]}