Question

How do you solve `abs(z+5)<3`?

Answer 1

See the entire solution process below:

Explanation:

The absolute value function transforms any negative or positive term into its positive form. Therefore, we must solve the term within the absolute value for both its positive and negative solution. And, because this is an equality we must solve it as a system of inequalities written as:

`-3 < z + 5 < 3`

Now, subtract `color(red)(5)` from each segment of the system to solve for `z` while keeping the system of inequalities balanced:

`-3 - color(red)(5) < z + 5 - color(red)(5) < 3 - color(red)(5)`

`-8 < z + 0 < -2`

`-8 < z < -2`

Answer 2

See below.

Explanation:

Just a thought but this looks like a complex number problem, in which case `abs( z+ 5) color(red)(=) 3` is the equation of a circle of radius 3 centred on `- 5 + 0 i`

If so, it follows that the inequality is the inside of that circle in the complex plane.