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6x^3 - 9x^2<0

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3 Answers

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).
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.

Comments

check x = 0 in original equation

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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
 

Comments

Sorry for the competing answer, Jim.  We posted in parallel.  ;-)
No problem. That seems to happen a lot.
jim

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