Question

How do you solve `-17< 4- 3n \leq 15`?

Answer 1

See a solution process below:

Explanation:

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

`-17 - color(red)(4) < 4 - color(red)(4) - 3n <= 15 - color(red)(4)`

`-21 < 0 - 3n <= 11`

`-21 < -3n <= 11`

Now, divide each segment by `color(blue(-3)` to solve for `n` while keeping the segment balanced. However, because we are dividing or multiplying inequalities by a negative number we must reverse the inequality operators:

`(-21)/color(blue)(-3) color(red)(>) (-3n)/color(blue)(-3) color(red)(>=) 11/color(blue)(-3)`

`7 color(red)(>) (color(red)(cancel(color(black)(-3)))n)/cancel(color(blue)(-3)) color(red)(>=) -11/3`

`7 color(red)(>) n color(red)(>=) -11/3`

Or

`n >= -11/3` and `n < 7`

Or, in interval notation:

`[-11/3, 7)`