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

Aug 19, 2015

$- 15 < z < - 8$

#### 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:
$- 15 < z < - 8$