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

Sep 26, 2017

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.

$- 4 \le 3 x - 5 \le 4$

First, add $\textcolor{red}{5}$ to each segment of the system of inequalities to isolate the $x$ term while keeping the system balanced:

$- 4 + \textcolor{red}{5} \le 3 x - 5 + \textcolor{red}{5} \le 4 + \textcolor{red}{5}$

$1 \le 3 x - 0 \le 9$

$1 \le 3 x \le 9$

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

$\frac{1}{\textcolor{red}{3}} \le \frac{3 x}{\textcolor{red}{3}} \le \frac{9}{\textcolor{red}{3}}$

$\frac{1}{3} \le \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} x}{\cancel{\textcolor{red}{3}}} \le 3$

$\frac{1}{3} \le x \le 3$

Or

$x \ge \frac{1}{3}$ and $x \le 3$

Or, in interval notation:

$\left[\frac{1}{3} , 3\right]$