Question

How do you solve and write the following in interval notation: `4< -z - 4 < 11`?

Answer 1

See a solution process below:

Explanation:

First, add `color(red)(4)` to each segment of the system of inequalities to isolate the `z` term while keeping the system balanced:

`4 + color(red)(4) < -z - 4 + color(red)(4) < 11 + color(red)(4)`

`8 < -z - 0 < 15`

`8 < -z < 15`

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

`color(blue)(-1) xx 8 color(red)(>) color(blue)(-1) xx -z color(red)(>) color(blue)(-1)xx 15`

`-8 color(red)(>) z color(red)(>) -15`

Or

`z < -8`; `z > -15`

Or, in interval notation:

`(-15, -8)`