Question

How do you solve and write the following in interval notation:`−12 <3 − 3x ≤ 15`?

Answer 1

See a solution process below:

Explanation:

First, subtract `color(red)(3)` from each segment of the system of inequalities to isolate the `x` term while keeping the system balanced:

`-color(red)(3) - 12 < -color(red)(3) + 3 - 3x <= -color(red)(3) + 15`

`-15 < 0 - 3x <= 12`

`-15 < -3x <= 12`

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

`(-15)/color(blue)(-3) color(red)(>) (-3x)/color(blue)(-3) color(red)(>=) 12/color(blue)(-3)`

`5 color(red)(>) (color(blue)(cancel(color(black)(-3)))x)/cancel(color(blue)(-3)) color(red)(>=) -4`

`5 color(red)(>) x color(red)(>=) -4`

Or

`x < 5` and `x >= -4`

Or, in interval notation:

`[-4, 5)`