6x^3 - 9x^2<0

Question

6x³ - 9x² <0

Solving for all values of x.

 

 

 

Answer 1

6x^3 - 9x^2 < 0

 

6x^2(x - 3/2) < 0

 

Left side has zeros at 0 and 3/2.

 

Left side is negative when x ≠ 0 and x < 3/2;

 

i.e., x ∈ (–∞,0) U (0,3/2).

Answer 2

6x³ - 9x² < 0

 

3x²(2x - 3) < 0

 

The 3x² term is always positive, so we must have:

 

(2x - 3) < 0

 

2x < 3

 

x < 3/2

 

On a number line, put a circle at x = 3/2 to indicate that 3/2 is not in the solution set, then darken in all values of x < 3/2.

Answer 3

Hi Cyn',

              You can write the inequality, by factoring out 3x², as follows

 

3x²(2x-3)<0  by inspection the roots are x=0 and x=3/2. so for x<0 the inequality is satisfied because 2x-3 <0 when x<0 and the inequality is satisfied when 0<x<3/2. When x>3/2 the function is positive so the inequality is not satisfied. 

 

Jim