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

Jan 28, 2018

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:

$- 2 \le r + 1 \le 2$

Subtract $\textcolor{red}{1}$ from each segment of the system of inequalities to solve for $r$ while keeping the system balanced:

$- 2 - \textcolor{red}{1} \le r + 1 - \textcolor{red}{1} \le 2 - \textcolor{red}{1}$

$- 3 \le r + 0 \le 1$

$- 3 \le r \le 1$

Or

$r \ge - 3$; $r \le 1$

Or, in interval notation

$\left[- 3 , 1\right]$

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: