The question I have is....

Using induction, verify the inequality

is greater than or equal to (1+nx)

for x is greater than or equal to -1 and n is greater than or equal to 1

The basis step was easy enough (as always)

However, I became stuck at the induction part...

Since the condition given is for BOTH x AND n, I tried doing induction for

x+1 and n+1

So I had the following...

Assuming the above statement to be true, prove

(2+x)^(n+1) is greater than or equal to 1+(n+1)(x+1)

I tried a lot of different ways, but couldn't solve it...

The closest thing I got to an answer was by doing the following...

(2+x)^(n+1)=(2+x)(2+x)^n

(2+x)(2+x)^n is greater than or equal to (1+x)^n+(n+x+1)

Since, (2+x)^n is greater than or equal to (1+x)^n I assumed that part of the equation to be true... leaving me to prove

(2+x) is greater than or equal to (n+x+1)/((1+x)^n)

which is easy enough to prove.... Since will ALWAYS be greater than 1 given the condition x is greater than or equal to -1 and n is greater than or equal to 1

But I'm pretty sure this is NOT the right way to prove this (since this is supposed to be proven using induction... and I feel like I am taking way too many assumptions)

Any hints/tips on how to solve an induction problem with two changing variables?