# Why is this inequality true?

I was solving an inequality and got stuck at this part: $$abc+abd+acd+bcd\le a^3+b^3+c^3+d^3$$ Why is this true? I think it has a similar solution as $ab+ba\le a^2+b^2$, because in both cases the left side is rearranged.

• $\{a,b,c,d\} \in \mathbb{R}$? – Maazul May 20 '13 at 13:20
• @Maazul: It has to be $a,b,c,d\geq 0$, because otherwise you could replace each value with its negative, and this would reverse the inequality. – Glen O May 20 '13 at 13:24
• oh yes, sorry, I forgot to say this. – D180 May 20 '13 at 13:36

Hint $a^3+b^3+c^3\ge 3abc$ for $a, b, c \ge 0$.
Just another track to the truth: Apply $3^\text{rd}$ degree AM-GM $$abc\:\leqslant\:\frac{a^3+b^3+c^3}{3}$$ to each summand on the LHS, followed by 'garbage collection'.