How do you solve and graph abs(4n+3)>=18?

Aug 22, 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.

We can solve this as a system of inequalities:

$- 18 \ge 4 n + 3 \ge 18$

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

$- 18 - \textcolor{red}{3} \ge 4 n + 3 - \textcolor{red}{3} \ge 18 - \textcolor{red}{3}$

$- 21 \ge 4 n + 0 \ge 15$

$- 21 \ge 4 n \ge 15$

Now, divide each segment by $\textcolor{red}{4}$ to solve for $n$ while keeping the system balanced:

$- \frac{21}{\textcolor{red}{4}} \ge \frac{4 n}{\textcolor{red}{4}} \ge \frac{15}{\textcolor{red}{4}}$

$- \frac{21}{4} \ge \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}} n}{\cancel{\textcolor{red}{4}}} \ge \frac{15}{4}$

$- \frac{21}{4} \ge n \ge \frac{15}{4}$

Or

$n \le - \frac{21}{4}$ and $n \ge \frac{15}{4}$

Or, in interval notation:

$\left(- \infty , - \frac{21}{4}\right]$ and $\left[\frac{15}{4} , + \infty\right)$

To graph, we will draw vertical lines at points $- \frac{21}{4}$ and $\frac{15}{4}$ on the horizontal axis.

Both line will be solid because there is a "or equal to" clause in both inequalities.

We will shade to the left and to the right of the two lines because of the "less than" and "greater than" clauses in the solutions.