Question

How do you solve and graph `abs(r+1)<=2`?

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. We can rewrite this problem as:

`-2 <= r + 1 <= 2`

Subtract `color(red)(1)` from each segment of the system of inequalities to solve for `r` while keeping the system balanced:

`-2 - color(red)(1) <= r + 1 - color(red)(1) <= 2 - color(red)(1)`

`-3 <= r + 0 <= 1`

`-3 <= r <= 1`

Or

`r >= -3`; `r <= 1`

Or, in interval notation

`[-3, 1]`

To graph this we will draw vertical lines at `-3` and `1` on the horizontal axis.

The lines will both be solid lines because their inequality operators both contain an "or equal to" clause. This indicates both `-3` and `1` are part of the solution set.

We will shade to the regional between the lines to show the solution set: