Question

How do you solve and write the following in interval notation: `2< -3x + 2<=14`?

Answer 1

See the entire solution process below:

Explanation:

First, subtract `color(red)(2)` from each segment of the system of inequalities to isolate the `x` term while keeping the system balanced:

`2 - color(red)(2) < -3x + 2 - color(red)(2) <= 14 - color(red)(2)`

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

`0 < -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 an inequality by a negative term we must reverse the inequality operators:

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

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

`0 > x >= -4`

Or

`x < 0` and `x >= -4`

In interval notation [-4, 0)