^{3}- 9x

^{2 }<0

6x^{3} - 9x^{2 }<0

Solving for all values of x.

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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).

6x^{3} - 9x^{2} < 0

3x^{2}(2x - 3) < 0

The 3x^{2} 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.

Hi Cyn',

You can write the inequality, by factoring out 3x^{2}, as follows

3x^{2}(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

Sorry for the competing answer, Jim. We posted in parallel. ;-)

No problem. That seems to happen a lot.

jim

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