Solve the inequality 12x^3 + 8x^2 <= 3x + 2. Justify your answer.

Answer 1

12x^3 + 8x^2 ≤ 3x + 2
12x3 + 8x^2 - 3x - 2 ≤ 0

after some trial and errors where I tried
x = ±1, ±2, etc, I found
x = ±1/2 and x = -2/3 give me zero

12x3 + 8x^2 - 3x - 2 ≤ 0
(2x-1)(2x+1)(3x+2) ≤ 0

knowing the behaviour of
y = 12x3 + 8x^2 - 3x - 2
and having x-intercepts of -2/3, -1/2 and +1/2

the solution would be
x ≤ -2/3 OR -1/2 ≤ x ≤ 1/2

confirmation:
http://www.wolframalpha.com/input/?i=plot+12x3+%2B+8x%5E2+-+3x+-+2+≤+0