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1 solution
Firstly simplify, using the remainder theorem to find roots of the equality:
x 3 − 2 x 2 − x + 2 = 0
Roots are found to be x = − 1 , 1 , 2 , so we can re-write the original equation:
x 3 − 2 x 2 − x + 2 < 0
as
( x + 1 ) ( x − 1 ) ( x − 2 ) < 0
There are a few options you can graph the function see where it is below the y-axis, you can trial and error solutions before, after and between roots, there are a number of choice here, choose which you feel most happy with.
Find that x < − 1 gives a negative solution, common sense as x 3 when x is a large negative number will dominate the equation and you'll never get anything other than a negative solution.
Similarly, x > 2 for large positive values of x gives a positive solution for the same reasons.
Now just check between the roots.
x = 0 gives a positive solution, so between the roots − 1 and 1 , the solutions will all be positive, try 1 . 5 as a solution between 1 and 2 , this gives a negative solution, so we have our answer as:
x < − 1 and 1 < x < 2