Question

How do you solve `x^2(4-x)(x+6)<0`?

Answer 1

The inequality is TRUE for values of x:
`x < -6" "` OR `" "x>4`

Explanation:

Since by solving for the values of x for each factor, we are going to have values `x=-6` and `x=0` and `x=4`

The intervals are `(-oo, -6)` and `(-6, 0)` and `(0, 4)` and `(4, +oo)`

Let us use test points for each interval

For `(-oo, -6)` , let us use `-7`

For `(-6, 0)` , let us use `-2`

For `(0, 4)` , let us use `+1`

For `(4, +oo)` , let us use `+5`

Let us do each test

At `x=-7" "`the value`" " " "x^2(4-x)(x+6)<0" "`TRUE
At `x=-2" "`the value`" " " "x^2(4-x)(x+6)<0" "`FALSE
At `x=+1" "`the value`" " " "x^2(4-x)(x+6)<0" "`FALSE
At `x=+5" "`the value`" " " "x^2(4-x)(x+6)<0" "`TRUE

Conclusion:

The inequality is TRUE for the following intervals
`(-oo, -6)` and `(4, +oo)`

OR

The inequality is TRUE for values of x:
`x < -6` OR `x>4`

God bless....I hope the explanation is useful.