# Homework inequality -- Show that (a+1)(b+1)(c+1)(d+1) < 8(abcd+1)

1. Mar 8, 2017

### ssd

1. The problem statement, all variables and given/known data
For a,b,c,d >1, Show that (a+1)(b+1)(c+1)(d+1) < 8(abcd+1)

2. Relevant equations
How to show this?

3. The attempt at a solution
I could show for two variables, (a+1)(b+1)<2(ab+1). Tried C-S, AM-GM inequalities in different form and variable transformations. But still no result. It's my daughters test question. Any help is appreciated.

Last edited: Mar 8, 2017
2. Mar 8, 2017

### ssd

Can this be done by induction?

3. Mar 8, 2017

### Buffu

(a+1)(b+1)<2(ab+1)
(d+1)(c+1)<2(dc+1)

(a+1)(b+1)(d+1)(c+1)< 4(ab+1)(dc+1)

Can you complete now ?

4. Mar 8, 2017

### issacnewton

ssd, Buffu has given a nice lead. Recognize that $ab>1$ and $cd>1$. So replace these in your expressions for $a$ and $b$

5. Mar 8, 2017

### ssd

Thanks a lot to both of you.