Solving Polynomial Inequalities of x3

The inequality to be solved is : x^3 <
x

x^3 < x

move all the
terms to the left

=> x^3 - x <
0

factor the left hand
side

=> x(x^2 - 1)<
0

=> x(x - 1)(x + 1) <
0

Now the left hand side is less than 0
if

• all the factors are
  negative

=> x < 0 , x - 1
< 0 and x + 1 < 0

=> x < 0 , x
< 1 and x < -1

x < -1 satisfies all
the conditions

• only one of the factors is less
  than 0

1. x < 0, x - 1 > 0 and x
+ 1> 0

=> x < 0 , x > 1 and x
> -1

no value of x can satisfy
this

2. x > 0 , x + 1 < 0 and x - 1 >
0

=> x > 0 , x < -1 and x >
1

no value of x can satisfy
this

3. x > 0, x + 1 > 0 and x - 1 <
0

=> x > 0, x > -1 and x <
1

this is satisfied by x > 0 and x <
1

So the values of x that satisfy x^3
< x lie in (-inf., -1)U(0 , 1)