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

Jul 23, 2017

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.

$- 8 \le 3 d - 1 \le 8$

First, add $\textcolor{red}{1}$ to each segment of the system of inequalities to isolate the $d$ term while keeping the system balanced:

$- 8 + \textcolor{red}{1} \le 3 d - 1 + \textcolor{red}{1} \le 8 + \textcolor{red}{1}$

$- 7 \le 3 d - 0 \le 9$

$- 7 \le 3 d \le 9$

Now, divide each segment by $\textcolor{red}{3}$ to solve for $d$ while keeping the system balanced:

$- \frac{7}{\textcolor{red}{3}} \le \frac{3 d}{\textcolor{red}{3}} \le \frac{9}{\textcolor{red}{3}}$

$- \frac{7}{3} \le \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} d}{\cancel{\textcolor{red}{3}}} \le 3$

$- \frac{7}{3} \le d \le 3$

Or

$d \ge - \frac{7}{3}$ and $d \le 3$

Or, in interval notation:

$\left[- \frac{7}{3} , 3\right]$