Question

How do you solve and graph `abs(3d-1)<=8`?

Answer 1

See a solution process below:

Explanation:

The absolute value function takes any negative or positive 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.

`-8 <= 3d - 1 <= 8`

First, add `color(red)(1)` to each segment of the system of inequalities to isolate the `d` term while keeping the system balanced:

`-8 + color(red)(1) <= 3d - 1 + color(red)(1) <= 8 + color(red)(1)`

`-7 <= 3d - 0 <= 9`

`-7 <= 3d <= 9`

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

`-7/color(red)(3) <= (3d)/color(red)(3) <= 9/color(red)(3)`

`-7/3 <= (color(red)(cancel(color(black)(3)))d)/cancel(color(red)(3)) <= 3`

`-7/3 <= d <= 3`

Or

`d >= -7/3` and `d <= 3`

Or, in interval notation:

`[-7/3, 3]`