1. ## Exponential Inequalities

Hi,

Could anyone help me solve the following inequalities algebraically?

2^x-3<x+1
(x+1)(x-2)(2^x)>0

My teacher mentioned something about Newton's method for solving the first one, but he wouldn't explain it to us. I did some research, but I still don't really understand, other than that I have to estimate a zero of 2^x-x-4>0 and use tangents to find a very close approximation of the actual root. As for the second, I can come up with the answer through reasoning and logic, but I would like to know how to solve it algebraically.

2. Originally Posted by scy
Hi,

Could anyone help me solve the following inequalities algebraically?

...
(x+1)(x-2)(2^x)>0

..., but I would like to know how to solve it algebraically.
The LHS of the inequality is a positive product.

Since $2^x>0$ for all $x \in \mathbb{R}$ the sign of the LHS depends on the product (x+1)(x-1).

A product of 2 factors is positive if the signs of the factors are equal. Thus you get:

$x+1>0\ \wedge\ x-1>0~\vee~x+1<0\ \wedge\ x-1<0$

$x>-1\ \wedge\ x>1~\vee~x<-1\ \wedge\ x<1$ Therefore

$\boxed{x>1~\vee~x<-1}$