# How do you graph and solve |2x+3|<=15?

Jan 20, 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.

$- 15 \le 2 x + 3 \le 15$

First, subtract $\textcolor{red}{3}$ from each segment of the system of equations to isolate the $x$ term while keeping the system balanced:

$- 15 - \textcolor{red}{3} \le 2 x + 3 - \textcolor{red}{3} \le 15 - \textcolor{red}{3}$

$- 18 \le 2 x + 0 \le 12$

$- 18 \le 2 x \le 12$

Now, divide each segment of the system by $\textcolor{red}{2}$ to solve for $x$ while keeping the system balanced:

$- \frac{18}{\textcolor{red}{2}} \le \frac{2 x}{\textcolor{red}{2}} \le \frac{12}{\textcolor{red}{2}}$

$- 9 \le \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} x}{\cancel{\textcolor{red}{2}}} \le 6$

$- 9 \le x \le 6$

Or

$x \ge - 9$; $x \le 6$

Or. in interval notation

$\left[- 9 , 6\right]$

To graph this we will draw vertical lines at $- 9$ and $6$ on the horizontal axis.

The lines will be a solid line because the inequality operators contain "or equal to" clauses.

We will shade between the lines to show the interval: