Question

How do you solve the inequality `abs(3x-5)<=4` and write your answer in interval notation?

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.

`-4 <= 3x - 5 <= 4`

First, add `color(red)(5)` to each segment of the system of inequalities to isolate the `x` term while keeping the system balanced:

`-4 + color(red)(5) <= 3x - 5 + color(red)(5) <= 4 + color(red)(5)`

`1 <= 3x - 0 <= 9`

`1 <= 3x <= 9`

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

`1/color(red)(3) <= (3x)/color(red)(3) <= 9/color(red)(3)`

`1/3 <= (color(red)(cancel(color(black)(3)))x)/cancel(color(red)(3)) <= 3`

`1/3 <= x <= 3`

Or

`x >= 1/3` and `x <= 3`

Or, in interval notation:

`[1/3, 3]`