12x^3-20X^2>-11x+2 help?

Let f(x) = 12x³ - 20x² + 11x - 2
f(2/3) = 12(2/3)³ - 20(2/3)² + 11(2/3) - 2 = 0
Hence, 3x - 2 is a factor of 12x³ - 20x² + 11x - 2.


12x³ - 20x² > -11x + 2
12x³ - 20x² + 11x - 2 > 0
(12x³ - 8x²) - (12x² - 8x) + (3x - 2) > 0
4x²(3x - 2) - 4x(12x² - 8x) + (3x - 2) > 0
(3x - 2)(4x² - 4x + 1) > 0
(3x - 2)(2x - 1)² > 0

For all real x, (2x - 1)² ≥ 0
Hence, 3x - 2 > 0
x > 2/3

12x^3-20X^2>-11x+2 ---------------(1)

Consider the equation:
12x^3-20x^2 = -11x+2
12x^3-20x^2+11x-2 = 0
http://www.wolframalpha.com/input/?i=graph+12x%5E3-20x%5E2%2B11x-2
From the graph, x =1/2 is a solution

.........12x^2-14x+4
---- ---- --- - -- - -- - - - - -- -
x-1/2) 12x^3-20x^2+11x-2
.........12x^3-6x^2
------ ---------- ---------
..... ... ... ...-14x^2+11x
..... .... .......-14x^2+7x
---------- ----------- ----------
..... .... ... . .. .. . .....4x-2
.... ..... .. .. . .. . . .. .4x-2
------------ -------------- ----------
.......... ...... .. .. ......0

12x^3-20x^2+11x-2 = 0
(x-1/2)(12x^2-14x+4) = 0
12x^2-14x+4
= 12x^2-6x-8x+4
= 3x(4x-2) -2(4x-2)
=(3x-2)(4x-2)

(x-1/2)(12x^2-14x+4) = 0
(x-1/2) (3x-2)(4x-2) = 0
x=1/2
x=2/3
x=1/2

Plot the points on a real line.
--------- --------- --------- --------
-∞........1/2.........2/3...... ∞

Consider the intervals (-∞, 1/2), (1/2, 2/3) , (2/3, ∞)
From each interval, choose any one point and test inequality (1)

(-∞, 1/2) : choose x=0
12x^3-20x^2>-11x+2
0 > 2 (false)

(1/2, 2/3) : choose x = 7/12
12x^3-20x^2>-11x+2
-4.42 > -4.41 (false)

(2/3, ∞) : choose x=1
12x^3-20x^2>-11x+2
-8 > -9 (true)

The solution is 2/3 < x < ∞
or
x > 2/3

Same as 12x^3-20x^2-(-11x+2)>0, i.e. 12x^3-20x^2+11x-2 > 0, factor

(3x-2)(-1+2x)^2 > 0 second factor never negative, so same as
3x-2>0, i.e.
x>2/3