Inequality: $\left|x^3-y^3\right|<|x|^3+|y|^3$

Question

Could anyone show me why

$$\left|x^3-y^3\right|<|x|^3+|y|^3$$
for all real numbers (x,y) except 0?

I'm thinking of whether of how to remove the modulus sign on the left hand side of the equation maybe using a distance formula. Should I raise the left hand side to the power of 2 to remove the modulus?

Answer 1

Note that $$|x^3 - y^3| = |x^3 + (-y^3)| \leq |x^3| + |(-y^3)| = |x^3| + |y^3|$$

The $\leq$ part comes from the triangle inequality ($|x+y| \leq |x| + |y|$)