Does |x^3 + x|=|(-x)^3 - x|?

Question

Does |x^3 + x|=|(-x)^3 - x|?

Answer 1

Yes, for example

 

let x = 3, then the left side equals 30 and the right side equals |-27-3| which also equals 30.

 

If x = -3, then left side is |-27-3| = 30 and right side = 27+3 = 30.

 

Answer is true for all real numbers

 

Answer 2

The right side of the inequality can be equivalently written as:

 

|(-1)(x^3) + (-1)(x)|

 

Which means we can pull the -1 out:

 

|(-1)(x^3 + x)|

 

Because of the absolute value signs, we can just eliminate the -1 entirely, yielding:

 

|x^3 + x|

 

Which is identical to the left side of the inequality. To check your work, try plugging in a few example values for x- I would try a negative number, a positive number, and 0!