Question 1141145: solve: |x|^(3x^2-4x-4) > 1 ,x ∈ R
Answer by greenestamps(13209)   (Show Source):
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Equality is achieved for |x|=1 or for 3x^2-4x-4=(3x+2)(x-2)=0. So the value of the expression is EQUAL to 1 for x = -1 and x = 1, and for x = -2/3 and x = 2.
Left to right, then, the x values where the expression is equal to 1 are -1, -2/3, 1, and 2. Determine the solution set for the inequality by considering values of x in each interval of the number line determined by those 4 values.
(1) x < -1 --> |x|>1 and 3x^2-4x-4 > 0 --> the expression value is greater than 1
(2) -1 < x < -2/3 --> |x|<1 and 3x^2-4x-4 > 0 --> the expression value is less than 1
(3) -2/3 < x < 1 --> |x|<1 and 3x^2-4x-4 < 0 --> the expression value is greater than 1
(4) 1 < x < 2 --> |x| > 1 and 3x^2-4x-4 < 0 --> the expression value is less than 1
(5) x > 2 -- |x| > 1 and 3x^2-4x-4 > 0 --> the expression value is greater than 1
ANSWER: The expression value is greater than 1 on (-infinity,-1), (-2/3,1), and (2,infinity)
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