Question

How do you solve `4< -z -4 < 11`?

Answer 1

`-15 < z < -8`

Explanation:

There are two inequalities here.
Firstly, let's solve them. Secondly, we will combine them into a resulting inequality for `z`.

1. `4 < -z - 4`
   To solve this inequality for `z`, add `z` to both sides of equation and then subtract `4` from both sides.
   The first transformation will bring positive `z` to the left side instead of negative in the right, getting
   `z+4 < z-z-4`
   `z+4 < -4`
   The second transformation will get rid of `4` on the left:
   `z+4-4 < -4-4`
   `z < -8`

2. `-z-4 < 11`
   To solve this inequality for `z`, add `z` to both sides of equation and then subtract `11` from both sides.
   The first transformation will bring positive `z` to the right side instead of negative in the left, getting
   `z-z-4 < z+11`
   `-4 < z+11`
   The second transformation will get rid of `11` on the right:
   `-4-11 < z+11-11`
   `-15 < z` or, equivalently, `z > -15`

So, we have two conditions on `z`:
`z < -8` and `z > -15`
We can combine them into one:
`-15 < z < -8`