Question

How do you solve and graph `abs(3t+6)<9`?

Answer 1

See a solution process below:

Explanation:

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

`-9 < 3t + 6 < 9`

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

`-9 - color(red)(6) < 3t + 6 - color(red)(6) < 9 - color(red)(6)`

`-15 < 3t + 0 < 3`

`-15 < 3t < 3`

Now, divide each segment by `color(red)(3)` to solve for `t` while keeping the system balanced:

`-15/color(red)(3) < (3t)/color(red)(3) < 3/color(red)(3)`

`-5 < (color(red)(cancel(color(black)(3)))t)/cancel(color(red)(3)) < 1`

`-5 < t < 1`

Or

`t > -5` and `t < 1`

Or, in interval notation:

`(-5, 1)`

To graph this we will draw a vertical lines at `-5` and `1` on the horizontal axis.

The lines will be dashed lines because the inequality operators do not contain an "or equal to" clause.

We will shade between the two lines to represent the solution interval: