Simple inequalities Suppose $l,t\in[0,1]$ and $l+t\leq1$ I want to prove $1+l+t>6lt$. When $t=0$ or $l=0$, it is trivial, so I started with $l,t\neq0$ but I couldn't reach anywhere. I don't have time to write in detail what I have already tried, but I tried to manipulate $(l-t)^2$ mostly. Anyway, if anyone help me with the proof that would be great. Many thanks!

By AM-GM,  $4lt \le (l+t)^2\le 1$.  
Similarly, $2 lt \le 2\sqrt{lt} \le l+t$.  Add these... and note equality is not possible for all the inequalities used simultaneously.
