# How do you solve the inequality abs(2x-3)>=5?

Jul 25, 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.

$- 5 \ge 2 x - 3 \ge 5$

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

$- 5 + \textcolor{red}{3} \ge 2 x - 3 + \textcolor{red}{3} \ge 5 + \textcolor{red}{3}$

$- 2 \ge 2 x - 0 \ge 8$

$- 2 \ge 2 x \ge 8$

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

$- \frac{2}{\textcolor{red}{2}} \ge \frac{2 x}{\textcolor{red}{2}} \ge \frac{8}{\textcolor{red}{2}}$

$- 1 \ge \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} x}{\cancel{\textcolor{red}{2}}} \ge 4$

$- 1 \ge x \ge 4$

Or

$x \le - 1$; $x \ge 4$

Or, in interval notation:

(-oo, -1]; [4, +oo)