Question

How do you solve and graph `-5<=-n-6<=0`?

Answer 1

See a solution process below:

Explanation:

First, add `color(red)(6)` to each segment of the system of inequalities to isolate the `n` term while keeping the system balanced:

`-5 + color(red)(6) <= -n - 6 + color(red)(6) <= 0 + color(red)(6)`

`1 <= -n - 0 <= 6`

`1 <= -n <= 6`

Now, multiply each segment by `color(blue)(-1)` to solve for `n` while keeping the system balanced. However, because we are multiplying or dividing inequalities by a negative number we need to reverse the inequality operators:

`color(blue)(-1) * 1 color(red)(>=) color(blue)(-1) * -n color(red)(>=) color(blue)(-1) * 6`

`-1 color(red)(>=) n color(red)(>=) -6`

Or

`n <= -1`; `n >= -6`

Or, in interval notation:

`[-6, -1]`