How do you solve x^3 − 2x > 1 ?

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How do you solve x^ 3 − 2x > 1 ?
How do you solve x^3 − 2x > 1 ?

How do you solve x^3 − 2x > 1 ?

Explanation:

First subtract 1 from both sides to get x3−2x−1>0.

Notice that x=−1 is a solution of x3−2x−1=0, so (x+1) is a factor of x3−2x−1. The other factor is x2−x−1.

x3−2x−1=(x2−x−1)(x+1).

Solve x2−x−1=0 using the quadratic formula

x=−b±√b2−4ac2a

=1±√1−4⋅1⋅(−1)2⋅1

=1±√52

So x3−2x−1 will cross the x-axis at x=−1, x=1−√52 and x=1+√52.

Since the dominant term is x3, the function x3−2x−1 will be negative for x<−1, positive for −1<x<1−√52, negative for 1−√52<x<1+√52 and positive for x>1+√52.

So the solution to the original problem is:

−1<x<1−√52 or x>1+√52.

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