Can someone solve this inequality

I have a question regarding inequalities. No matter which way I do it, I end up getting different answers.
The first problem is:
5 - | 2x + 4 | ≤ 3

The answer I got was:
x ≥ 3 and x ≥ 6
[ 3 , ∞ ] ⋃ [ 6 , ∞ ]

I tried it a different way and got:
x ≥ - 1 and x ≤ - 2
[ -1 , ∞ ] ⋃ [ - 2 , ∞ ]

I don't understand what I did wrong. I'll provide my work under additional information to reduce the space spent on here.

On the second problem, I got halfway and gave up.
The problem is:
x / x + 1 > 4x

or written differently:
x
------- > 4x
x + 1

5 - | 2x + 4 | ≤ 3 => 5-3<=|2x+4| => 2<=|2x+4|.
If 2x+4>=0 this is equivalent to 2<=2x+4 => -2<=2x => -1<=x
or x>=-1.
If 2x+4<0mthis is equivalent to 2<=-(2x+4)=>2<=-2x-4=> 6<=-2x
=> 6/(-2)>=x=>-3>=x or x<=-3.

Hence the answer is (-infinity,-3] Union [-1, +infinity).

Hope that helps.