Solve the inequality 4< -z -4 < 11?

4 < - (z + 4) < 11

-(z + 4) < 11
z + 4 > - 11
z > - 15

4 < - (z + 4)
- 4 > (z + 4)
- 8 > z

{ z : - 15 < z < - 8 }

4<-z-4<11
so you have two inequalities which are:
4<-z-4 ,,,,(1) and -z-4<11......(2)
start with eqn (1)
4<-z-4
8<-z
therefore
z<-8

and (2) -z-4<11
-z<15
z>-15 so therefore: -15<z<-8

4 < -z - 4 < 11
4 + 4 < -z < 11 + 4
8 < -z < 15
8/-1 > z > 15/-1
-8 > z > -15

4< -z -4 < 11
Add 4 to both sides
8 < -z < 15
Divide both sides by -1 remembering
to reverse the direction of the inequality
-8 > z > -15
and that is the answer. It can
be written
-15 < z < -8 . Put back in
set builder notation it would be {z|-15 < z < -8}
or {z|-8> z > -15} which is the first choice
in your question.
The second choice in your question makes
no sense because
-8< z < -15 means z < -15 and -8 < z which means z> -8
and one would have to wonder how in the (we'll say heck) z could be less than -15 while being greater than -8. Heavens (or Heck! should we say the other place) No!

1) add +4 to the two sides
4+4< -z <11+4
2) simplefy
8 < - z < 15
3) divide by -1
-8 > z > -15
the answer
z = -9 ,or, -10 ,or, -11,or, ...... -14.

4 < -z - 4 < 11
the equation can be form into two...
4 < -z - 4
-z > 8
z < -8

-z - 4 < 11
-z < 15
z > -15

-15 < z < -8

z = < -15, -8 >

ie -15 -14 -13 -12 -11 -10 -9 -8 but without -15 and -8

provided Z is a whole number

+4 to all
multiply by -1 (causes sign change)

-15< z < -8

-15 < z < -8

8<-z<15
or
-8>z>-15

i think...