How Do You Solve Complex Exponential Inequalities?

[enibaraliu]

Homework Statement

1) 5^x+4-5^x-1<24
2) 2^x+2-1>=2^2x+3
3) 2^1-x-2^x+1/2^x-1<0

Homework Equations

At the 2nd I do like this:
2^x+2-2^2x>=4 divide by 2
2^x+1-2^2x-1>=2
x+1-2x+1>=1
-x>=-1 divide by (-1)
x=<1
but when i put 0 in x the inequation is incorrect, what I have mistake, please tell me.
If you can help me in anothers inequations, thanks a lot.

The Attempt at a Solution

 

[enibaraliu]

Please can you help for this?

 

[enibaraliu]

I need the help urgently, please help!

 

[statdad]

Your simplifications with the exponents aren't correct. Try this:

$$\begin{align*} 2^{x+2} - 2^{2x} & \ge 4\\ 2^x \cdot 2^2 - 2^{2x} & \ge 4\\ 4\cdot 2^x - 2^{2x} -4 & \ge 0\\ 2^{2x} - 4\cdot 2^x + 4 & \le 0 \end{align*}$$

Now, for ease of reading, let

$$u = 2^x \Rightarrow u^2 = 2^{2x}$$

so the inequality is now

$$u^2 - 4u + 4 & \le 0$$